Math 30-1 Notes
Unit 6: Exponents and Logarithms
UNIT 6:
EXPONENTIAL
&
LOGARITHMIC
FUNCTIONS
pg. 1
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 1: Review of Exponents
A power is an expression of the form a n , where a is the base and n is the exponent.
The base represents the number that is multiplied repeatedly.
The exponent represents the number of times the base is multiplied.
For example, 23  2  2  2 .
exponent
base
Exponent Laws
For the following exponent laws, a, b, m, and n are rational numbers or variables.
Exponent Law
Explanation
The
product
of
powers
with the same base
 Product of Powers:
can be simplified to a single power with
a m  a n  a m n
the same base by adding the exponents.
The quotient of powers with the same base
 Quotient of Powers:
can be simplified to a single power with
am
m
n
mn
the same base by subtracting the
a  a  n  a ,a  0
a
exponents.
 Power of a Power:
a 
m n
 a mn
 Power of a Product or Quotient:
m
a b
 ab 
   m ,c  0
c
 c 
m m
 Zero Exponent:
a0  1, a  0
 Negative Exponent:
a
m
1
 m
a
 Rational Exponent:
m
an 
 
n
a
m
Example
x 2  x3  x5
x5
x  x  3  x2
x
5
A power raised to an exponent can be
expressed as a single power with the same
base by multiplying the exponents.
3
( x3 ) 2  x 6
A product or quotient raised to a power can
be represented with each term in the
product or quotient raised to the power.
2
 3x  9 x 2
   2
y
 y 
Any non-zero base raised to the exponent
of zero is equal to one.
A power with a negative exponent can be
written as the reciprocal of the power with
a positive exponent.
A power with a rational exponent can be
written as a radical, by placing the base
under a radical sign with the same index as
the denominator of the exponent, and
raising it to numerator of the exponent.
pg. 2
power
x0  1
 3x 
3

3
1
 3x 
3

1
27 x3
 4x 
 2 x 
 4x 2 
3
3
 8 x3
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Example
Simplify the following expressions. Express the final answer with only positive
exponents.
24m5 p 3q 4
5 2 2
2 4
a)  3x y   2 x y 
b)
4m4 p 2 q 2
 5x3 y 
c)  1 4 
x y 
Example
1
2
12 x 2
d)
3x
Evaluate the following without using technology.
1
2
a) 49 2
b) 32 5
c) 25

 8 
d) 

 125 
3
2
pg. 3

2
3
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Covert each of the following to the base indicated.
a) 92 x to base 3
b) 1252 x to base 5
c) 8 16 to base 2
 16 
d)  
 81 
x
Example
to base
2
3
Express as a single power.
 9  3 
2 x 1
a)
x 5
27
x4
b)
x 1
HOMEWORK:
Page 162, #1 – 5, 7 – 9
;
pg. 4
 2x   12 
x 7
Page 166, #1
 2
4 x 6
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 2: Exponential Functions and Transformations
Activity
1. Consider the exponential function y  2 x .
a) Use a table of values to sketch the graph of y  2 x .
x
3
2
y
1
0
1
2
10
3
y
b) Determine the following characteristics.
i. domain and range
ii. x- and y-intercepts
iii. equation of the horizontal asymptote
x
1
2. Consider the exponential function y    .
2
x
-4
x
1
a) Use a table of values to sketch the graph of y    .
2
0
1
2
3
x
3
2
1
4
-2
y
10
y
b) Determine the following characteristics.
i. domain and range
ii. x- and y-intercepts
iii. equation of the horizontal asymptote
x
1
3. How are the graphs of y  2 and y    related? Explain.
2
x
4. a) Predict what the graph of y  c x will look like if c = 1.
b) Predict what the graph of y  c x will look like if c < 0.
pg. 5
x
-4
4
-2
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Exponential Functions
An exponential function is of the form y  ab x , where b  0 and x is the exponent.
If b  1 , then the function is increasing and used to model exponential growth.
If 0  b  1 , then the function is decreasing and used to model exponential decay.
There are no x-intercepts because the x-axis  y  0  is the horizontal asymptote.
The domain is  x|x 
 and the range is  y|y  0, y   , if
Exponential Decay
Exponential Growth
Example
Describe how the graph of the second function compares to the graph of the first
function.
a) y  4 , y  2  4 
x
x 2
b) y  2 , y  4  2
x
x
5
The following transformations are applied to the base function y  3x . State the
equation of the transformed image.
a reflection in the x-axis
a vertical stretch by a factor of 5 about the x-axis
a horizontal stretch by a factor of 2 about the y-axis
a horizontal translation 1 unit right
a vertical translation 7 units down
Example
●
●
●
●
●
a  0.
pg. 6
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
a) Describe, in order, a series of transformations that could be applied to the graph of
y  2 x to produce the graph of y  16  2 x3  4 . Include lines about which
stretches and reflections occur.
b) Determine the x- and y-intercepts of the transformed function.
c) State the equation of any asymptotes.
d) State the domain and range.
HOMEWORK:
Page 176, #1 – 6, 8 – 11
pg. 7
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 3: Understanding Logarithms
Properties of the Inverse Function
f and f 1 “cancel” each other, since they are inverses (opposites) of each other.
f 1  f ( x)   x
ie.
f  f 1 ( x)   x
and
Logarithmic Functions
A log function is the inverse of the exponential function y  b x .
If follows that
logb  b x   x ,
blog b x  x
,
x
logb x  y
iff
by  x
x>0
The most widely used logarithms are called common logarithms.
Common logarithms are logarithms with base 10.
log x  log10 x
A common log is the inverse of 10 x . Thus, it follows that:
log 10x  x
Example
and
10log x  x
Convert each of the following from logarithmic form to exponential form.
a) log 7 x  4
b) log5 15  y
c) m  logt B
d) 5  4logb 6
pg. 8
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Convert each of the following to exponential form ( y  ab x ).
Example
 y
a) log 7    x
3
Example
b) log5  7 y   x
Convert each of the following from exponential form to logarithmic form.
1
8
a) 43  x
b) 23 
d) 2 x  4  a5
e) ed  f
3
f) y  (10) x
2
a) log 6 36
b) log 7 1
c) log1000
d) log 1 32
e) log 2 8
f) 5log1  6log10
Example
c) y  2(3x )
Evaluate.
2
pg. 9
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Determine the value of x.
a) log5 x  3
b) log16 x 
3
4
c) log x 49  2
d) log x 9 
2
3
Example
Given log 2 14 .
a) Between what two whole numbers is log 2 14 ? Find the numbers without using technology.
b) Use technology to find the approximate value of log 2 14 , to the nearest hundredth.
Change of Base Formula
log c x 
log d x
log d c
Using the change of base formula, for base d = 10,
log10 14
we obtain the approximation
.
log10 2
HOMEWORK:
Page 186, #3 – 7, 9 – 11, 15
pg. 10
;
Page 196, #1, 3, 4, 7 – 10, 13
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 4: Logarithmic Functions and Transformations
Activity
1. a) Complete the table of values shown below, for the exponential function y  10 x . Round the
y-values values to the nearest hundredth.
0
0.5
x
1.5
1.0
0.5
y
b) The line y = x is graphed below. On the same coordinate axes sketch the graph of y  10 x .
y
y= x
3
2
1
-3
-2
1
-1
x
2
3
-1
-2
-3
2. a) Complete the table of values for the inverse of y  10 x . Round the x-values values to the
nearest hundredth.
x
0
0.5
y
1.5
1.0
0.5
b) On the coordinate axes shown above, sketch the graph of the inverse.
c) How are the graphs of y  10 x and its inverse related?
d) Is the inverse a function? Explain.
e) State the equation of the inverse.
pg. 11
Math 30-1 Notes
Unit 6: Exponents and Logarithms
The exponential function y  b x , where b  0 and b  1, and the logarithmic function y  logb x
are inverses of each other. Algebraically show this relationship.
Graphs of Logarithmic Functions
If b > 1:
If 0 < b < 1:
e.g.
e.g.
5
5
(2, 4)
4
4
(2, 4)
3
3
2
2
(4, 2)
(0, 1)
(0, 1) 1
1
0
0
-5
-4
-3
-2
-1
0
-1
1
2
(1, 0)
3
4
5
-5
-4
-3
-2
-1
1
(1, 0)
2
3
4
-2
-2
y=x
0
-1
-3
y=x
-3
-4
-4
-5
-5
From both graphs, we see:
For y  logb x :
For y = bx :
Domain: x 
Range:
Domain: x  0
y0
Range:
Asymptote: y  0
y 
Asymptote: x  0
pg. 12
Can’t take
the log of a
negative
number or 0
5
(4, 2)
Math 30-1 Notes
Unit 6: Exponents and Logarithms
x
Example
1
Given f ( x)    .
2
a) Sketch the graph of f ( x) .
x
3
2
1
y
0
1
2
3
8
y
b) State the equation of the inverse.
x
-8
c) Sketch the graph of the inverse.
x
0
y
3
2
1
8
-8
1
2
3
d) State the domain and range of the inverse and the equation of any asymptotes.
e) Algebraically determine any x- and y-intercepts of the inverse.
Example
Determine the inverse function of the following.
a) y  10 x1  4
b) y  log 1  x  5  6
4
pg. 13
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
a) Describe, in order, a series of transformations that could be applied to the graph of
y  log 2 x to produce the graph of y  5log 2  x  4  . Include lines about which
stretches and reflections occur.
b) State the equation of the vertical asymptote of y  5log 2  x  4  .
c) State the domain and range of y  5log 2  x  4  .
d) Algebraically determine the x- and y-intercepts of y  5log 2  x  4  .
Example
The graph of y = g(x) can be generated by stretching the
graph of y  log 4 x . Write an equation of y = g(x).
y = g(x)
pg. 14
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Welding is the most common way to permanently join metal parts together. Welders
wear a helmet fitted with a filter shade to protect their eyes from the intense light and
radiation produced by a welding light. The filter shade number, N, is defined by the
7   log T 
function N 
 1 , where T is the fraction of visible light that passes through
3
the filter. Shade numbers range from 2 to 14, with a lens shade number of 14 allowing
the least amount of light.
a) Describe how the function is transformed from N = logT.
b) The correct filter shade depends on the type of welding. A shade number of 6 is suggested
for Oxy-Fuel cutting when the plate thickness is greater than 150 mm. What fraction of
visible light is allowed to the welder, as a percent to the nearest hundredth?
HOMEWORK:
Page 186, #1, 8, 13, 14
;
pg. 15
Worksheet
Math 30-1 Notes
Unit 6: Exponents and Logarithms
HOMEWORK
1. Given f  x   3x .
10
y
a) State the equation of the inverse, f 1  x  .
b) Sketch the graph of f(x) and its inverse.
c) Identify the following characteristics of the inverse graph.
i. the domain and range
ii. the x-intercept, if it exists
iii. y-intercept, if it exists
iv. the equation of the asymptote
x
10
-10
-10
2. Given f  x   log 1 x .
4
a) State the equation of the inverse, f 1  x  .
10
y
b) Sketch the graph of y = f(x) and its inverse.
c) Identify the following characteristics of the inverse graph.
i. the domain and range
ii. the x-intercept, if it exists
iii. y-intercept, if it exists
iv. the equation of the asymptote
x
10
-10
-10
1

3. The point  ,  3  is on the graph of the logarithmic function f  x   logc x , and the point
8

 4, k  is on the graph of the inverse, y  f 1  x  . Determine the value of k.
4. State the equation of the inverse, f 1  x  , for each of the following.
b) f  x   5x4
a) f  x   log3 x  2
pg. 16
Math 30-1 Notes
Unit 6: Exponents and Logarithms
5. In each case, the graph of y = g(x) is a stretch of the graph of y = f(x).
Write the equation of y = g(x).
y
y
a)
b)
4
4
f .( x ) = log x
2
f .( x ) = log x
4
y = g .( x )
y = g .( x )
x
10
-2
x
10
-2
-4
-4
6. Each of the following functions is in the form y  a log6  b  x  h    k . For each function,
i. describe the transformations that correspond to the parameters a, b, h and k.
ii. state the domain and range
iii. algebraically determine the y-intercept
iv. algebraically determine the x-intercept.
a) y  log6  4  x  9  
b) y  3log6  x  1  6
7. The graph of y  log3 x has been transformed to y  a log3  b  x  h    k . Find the values of a,
b, h and k for each of the following transformations, and state the equation of the transformed
image.
1
a) a reflection in the x-axis, a horizontal stretch by a factor of about the y-axis, a translation
5
of 6 units left and a translation of 1 unit up
3
about the x-axis, a horizontal stretch by a factor of 4 about
4
the y-axis, a reflection about the y-axis, a translation of 2 units right and a translation of 5
units down
b) a vertical stretch by a factor of
8. a) Only a vertical translation has been applied to the graph of y  log3 x so that the graph of the
transformed image passes through the point (9, 4). Find the equation of the transformed
image.
b) Only a horizontal stretch has been applied to the graph of y  log 2 x so that the graph of the
transformed image passes through the point (8, 1). Find the equation of the transformed
image.
pg. 17
Math 30-1 Notes
Unit 6: Exponents and Logarithms
9. According to the Ehrenberg relation, the average measurements of heights, h, in centimetres,
and weights, w, in kilograms of children between the ages of 5 and 13 are related by the
function log w  0.008h  0.4 .
a) Predict the height of an 10 year old child weighing 60 kg, to the nearest centimetre.
b) Predict the weight of 12 year old child with a height of 150 cm, to the nearest kilogram.
10. The graph of f  x   log8 x can also be described by the equation g  x   a log 2 x . Find the
value of a.
SOLUTIONS
10
1. a) f  x   log3 x
b) See graph on right.
c) i. domain:  x x  0, x 
y
x
y =3
1

; range:
y= x
y =log x
y y  
3
x
10
-10
ii. 1
iii. no y-intercept
iv. vertical asymptote x = 0, the y-axis
-10
x
1
2. a) f 1  x    
4
b) See graph on right.
c) i. domain:  x x 
y = 14
x
( )

; range:  y y  0, y 
10
y
y= x

ii. no x-intercept
iii. 1
iv. horizontal asymptote y = 0, the x-axis
x
10
y =log 1 x
-10
4
-10
3. 16
4. a) f 1  x   3x2
b) f 1  x   log5 x  4
1
5. a) g  x   log 2 x
3
1 
b) g  x   log 4  x 
2 
pg. 18
Math 30-1 Notes
Unit 6: Exponents and Logarithms
6. a) i. a = 1 corresponds to no vertical stretch,
1
about the y-axis,
4
h = 9 corresponds to a horizontal translation of 9 units left, and
k = 0 corresponds to no vertical translation.
b = 4 corresponds to a horizontal stretch by a factor of
ii. domain  x x  9, x 
 and range  y y  
iii. 2
iv. 8.75
b) i. a = 3 corresponds to a reflection in the x-axis and a vertical stretch by a factor of 3 about
the x-axis
b = 1 corresponds to a no horizontal stretch,
h = 1 corresponds to a horizontal translation of 1 unit left, and
k = 6 corresponds to a vertical translation of 6 units down.
ii. domain  x x  1, x 
 and range  y y  
iii. 6
iv. 
35
36
7. a) a = 1, b = 5, h = 6, k = 1
This represents the equation y   log3  5  x  6    1 .
b) a =
3
1
, b =  , h = 2, k = 5
4
4
3
 1

This represents the equation y  log3    x  2    5 .
4
 4

8. a) y  log3 x  6
1 
b) y  log 2  x 
4 
9. a) 172 cm
b) 40 kg
10. a 
1
3
pg. 19
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 5: Logarithm Laws
Since logarithms are the inverses of exponents, the laws of logarithms are related to the exponent
laws. In the following table, a, b, m, n, M and N are rational or variables, where M , N , b  0 .
Law
Exponents
Logarithms
Product
a m  a n  a m n
logb  MN   logb M  logb N
Quotient
am
 a mn
n
a
Power
Example
a 
m
n
 a mn
M
logb 
N

  logb M  logb N

logb M N  N logb M
Description
The logarithm of a product of
numbers is the sum of the logarithms
of the numbers.
The logarithm of a quotient of
numbers is the difference of the
logarithms of the dividend and
divisor.
The logarithm of a power of a
number is the exponent times the
logarithm of the number.
Evaluate the following.
a) log 2 12  log 2 3
b) log6 9  log6 8  log6 2
c) 3log x  log x3
d) 2log5  2log 2
273
e) log 3 4
81
1


f) 2log 2 12   log 2 6  log 2 27 
3


pg. 20
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Example
Write each as a single logarithm in simplest form.
log3 x
a) 3log 2 x
b)
2
1
c) 2log7 x  log7 y  log7 z
3
d) log5 x 
1
 log5 y  log5 z 
2
Example
Write each expression in terms of the individual logarithms of x, y and z.
 xy 
a) log5  
b) log 4 x6
z
 
 
 x3 
d) log 7 

y z
 1 
c) log 7  2 
x 
Example
If log 2 x  a , determine an expression for log 2 16x 2 in terms of a.
HOMEWORK:
Page 202, #1 – 8, 10 – 13
;
pg. 21
Page 209, #1 – 5, 7 – 11
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 6: Solving Logarithmic Equations
A logarithmic equation is an equation containing the logarithm of a variable.
Example
a) Solve the following logarithmic equations.
ii) 2log x  log16
i) log x2  log16
b) Explain why the solutions to the logarithmic equations are not the same even
though 2log x  log16 can be written as log x 2 .
Solving Log Equations Algebraically
● Method #1:
If all terms contain the log function, write each side as a single log. Then, raise
each side as a power of the common base (inverse).
ie. If logb L  logb R , then blogb L  blogb R .
● Method #2:
If the equation has both logs and constants (numbers), move all the log terms to
one side and the constants to the other side. Then, raise each side as a power of
the common base (inverse).
ie. If logb L  R , then blogb L  b R .
**Verify all solutions by substituting the solution(s) back into the original equation- it is not
possible to take the log of zero or of a negative number.
pg. 22
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Algebraically solve the following equations.
a) log  6 x  2   log  x  1  log8
b) log 2 3 x2  30 x  2
c) log 2  x  6   3  log 2  x  8
pg. 23
Left Side
Right Side
Left Side
Right Side
Left Side
Right Side
Left Side
Right Side
Left Side
Right Side
Math 30-1 Notes
Unit 6: Exponents and Logarithms
d) log 2  log x (3x  4)   1
Example
Left Side
Right Side
Left Side
Right Side
Algebraically solve the equations below.
a) log x2  3log x  10
HOMEWORK:
b) (log x)2  3log x  0
Page 263, #1 – 7, 9 – 12
pg. 24
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 7: Solving Exponential Equations
An exponential equation is an equation where the exponent is a variable.
Solving Exponential Equations Algebraically
● Method #1:
● Method #2:
Example
2 x3
a) 5
Example
Change to a common base and make the exponents equal to each other.
Log both sides and use the power law to move the variable out of the exponent.
Using a common base, solve the following exponential equations.
5
7
 125 
c) 

 216 
b) 3  27 3
x

x
4
6
 
5
Given the equation 27 x  3x8 . Solve this equation by
a) changing the base.
b) taking the logarithm of both sides.
c) using the graphing calculator.
pg. 25
3 x 3
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Solve the following equations, expressing the solution as an exact value in the form
log M
and as a decimal, rounded to the nearest thousandth.
log N
a) 2(4 x )  16
b) 52 x1  75
c) 23 x1  5x
d) 2  3
HOMEWORK:
Page 167, #2ad, 3, 5, 6abd, 7 – 12
pg. 26
;
x 2
 7x
Page 236, #1 – 6, 9 – 11, 14, 15
Math 30-1 Notes
Unit 6: Exponents and Logarithms
LESSON 8: Applications of Exponential and Logarithmic Functions
Exponential and logarithmic functions can be applied to many real-world situations.
Below are a few application formulas that involve exponents and logarithms.
● Earthquake Magnitude:
 A
The Richter magnitude of an earthquake, M, is defined as M  log   , where A is the
 A0 
amplitude of the ground motion, measured by a sensitive seismometer, and A0 is the
amplitude, corrected for the distance to the actual earthquake, that would be expected for a
"standard" earthquake with an amplitude of 0.0001 cm and magnitude 0.
We can compare the earthquake magnitudes, M1 and M 2 by taking the difference of the
magnitudes.
A 
M 2  M1  log  2  ,
 A1 
where A1 and A2 are the amplitudes of the two earthquakes, and M 2  M1
● Sound Intensity:
I 
I 
The decibel levels of two sounds,  2 and 1 , are defined as  2  10log  2  and 1  10log  1  .
 I0 
 I0 
We can compare the intensities of the two sounds, I 2 and I1 , by taking the difference of
their decibel sound levels.
I 
β2  β1  10log  2  , where 2  1
 I1 
● Acidity:
The pH level of a solution is defined as pH   log  H   , where H  is the hydrogen ion
concentration in moles per litre.
We can compare the acidity of two solutions, A2 and A1, by taking the difference of their pH
levels, pH2 and pH1.
 H 
pH 2  pH1   log  2  ,
H 1
where H 1 and H  2 are the hydrogen ion concentrations of the two solutions, in moles per
litre, and pH 2  pH1 .
pg. 27
Math 30-1 Notes
Unit 6: Exponents and Logarithms
● Exponential Growth/Decay:
There are two general models for exponential growth/decay:
● A  A0 1  i  , where A is the final amount; A0 is the initial amount; i is the rate of
n
growth/decay per period, expressed as a decimal; and n is the number
of periods.
● A  A0 (r )
t
p
, where A is the final amount; A0 is the initial amount; r is the rate of
growth/decay; t is the total time; and p is the time for one period.
(formula sheet: y  ab
t
p
)
Example
An earthquake in Turkey in 1999 had a magnitude of 7.4 on the Richter scale. An
earthquake in Seattle in 1996 measured 5.3 on the Richter scale. How many times as
intense was the earthquake in Turkey as the one in Seattle?
Example
The pH scale is used in chemistry to measure the acidity or alkalinity of a solution.
The pH of a solution is defined as pH   log H   where H  is the hydrogen ion
concentration in moles per litre. The lower the pH, the more acidic the solution.
A common ingredient in many cola drinks is phosphoric acid. Coca-Cola Classic is 13
times as acidic as orange juice. If orange juice has a pH of 3.8, then what is the pH of
Coke Classic? Answer to the nearest tenth.
pg. 28
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
The intensity level, , of a sound in decibels (dB) is defined
 I 
as   10log   , where I is the intensity of the sound
 I0 
measured in watts per square metre (W/m2) and Io is
1012 W/m2 , corresponding to the faintest sound that can
be heard by a person of normal hearing.
a) Audiologists recommend that people should wear a hearing protector
if the sound level exceeds 85 dB. What is the intensity, in W/m2, of
such sounds?
b) The sound level of a chainsaw is about 85 dB. The maximum volume setting of a portable
media player with headphones is about 110 dB. To the nearest whole number, how many
times as intense as the sound of a chainsaw is the maximum volume setting of the portable
media player?
c) Sounds that are at most 100 000 times as intense as a whisper are considered safe, no matter
how long or how often you hear them. The sound level of a whisper is about 20 dB. What
sound level, in decibels, is considered safe no matter how long it lasts?
pg. 29
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
Anna plans to buy a $10 000 car when she earns her post-secondary degree in four
years. She invests $8000 in a term deposit that pays an annual percentage rate (APR)
of 6%, compounded semi-annually.
How long will it take for Anna to have enough money in her investment to buy the car?
Answer to the nearest tenth of a year.
Note:
Interest is a fee paid for the use of
money. In compound interest the
interest is paid at regular intervals and
added to the principal for the next
interest period.
 annually  once a year
 semi-annually  twice a year
 quarterly  4 times a year
 monthly  12 times year
 weekly  52 times a year
 daily  365 times a year
Example
The number of fish in a lake is increasing by 5% each year.
a) Write an equation to represent the number of fish present after t years. Use N0 to represent
the initial population and N(t) to represent the final population.
b) If there were 2500 fish present in June 2012, how many would you expect to be present in
June 2017?
c) How many years, to the nearest tenth, would it take for the fish population to increase to
double the number in June 2012?
pg. 30
Math 30-1 Notes
Example
Unit 6: Exponents and Logarithms
A laptop computer is purchased for $2 500. The value of the computer depreciates
30% per year.
a) If V is the value of the computer after t years, then what exponential equation could be used
to represent this situation?
b) After how many years will the laptop be worth less than $250? Answer to the nearest tenth
of a year.
A half-life is the length of time for an unstable element to decay to one half of its original mass.
Example
When an animal dies the amount of radioactive Carbon-14 (C-14) in its bones decays.
Archaeologists use this fact to determine the age of a fossil based on the amount of C14 remaining. The half-life of C-14 is 5 730 years.
a) If A is the percent of C-14 after t years, then what exponential equation could be used to
represent this situation?
b) The oldest bison bones unearthed at a site had 49.5% of the C-14 left. How old are the bones
when they were found, to the nearest year?
pg. 31
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Example
A sample of bacteria contained 387 cells. Twenty four hours later, the sample was
recounted and was found to contain 8012 cells. Determine the doubling period of the
bacteria to the nearest tenth of an hour.
Example
An annuity is a series of equal payments of money over equal intervals of time. The
total investment value of an annuity at the end of the time period is called the future
value. The formula for the future value of an annuity, FV, is
n
R 1  i   1

FV  
i
where n is the number of equal periodic payments of R dollars, and i is the interest rate
per compounding period. At the end of each quarter, Aaron makes a $625 payment
into a mutual fund that earns an annual percentage rate of 6%, compounded quarterly.
How long will it take Aaron’s annuity to be worth $1 000 000? Answer to the nearest
tenth of a year.
HOMEWORK:
Page 274, #1 – 5, 7, 11 ; Page 246, #3a, 5 – 9, 11
Page 253, #1, 3 – 5
;
Page 256, #8 – 15
pg. 32
Math 30-1 Notes
Unit 6: Exponents and Logarithms
Review (do on separate paper)
1. Solve the following exponential equations.
x
2 x1
a) 8
c)
2
5x 3
 
1
    32 
4
b) 4 9 x1  108
3
 45
d)
 3x  2  5  8
2. Convert the following to logarithmic form.
a) 34 
1
81
b)
ax
z
ay
c) a  bc d
3. Convert to exponential form with positive exponents.
1
 y
a) log r    h
b) log    x
m
4
4. Determine the exact value of the following logarithms using laws of logarithms.
a) log 4  log3 81
d)
log 36
log 6
b) 2
e)
 1 
log 3  
 27 
c) 5log5 1002log5 2
log x 6
log x 2
f) log 4 16 3 4  log3 3 27
5. Determine the value of the following logarithms.
a) If log3 4  x, then determine log3 4  2log3 16 .
b
b) If log a b  4.5,log a c  3.7, then determine log a   .
c
c) If log b 3  x,log b 2  y , then determine logb 48 .
1
d) If logb A  M , then determine logb 2 .
A
1 
e) If then log3 x  15 determine log3  x  .
3 
pg. 33
Math 30-1 Notes
Unit 6: Exponents and Logarithms
b g
6. a) Determine the x-intercept of the graph of y  log2 x  2 .
bg
b) The graph of the exponential function f x  b
bg
bg
x 2
, b  0, b  1 intersects the y-axis at ____.
c) Draw the graphs of f x  3x and f x  log3 x and state their domain and range.
d) The graph of f  x   log3 x passes through the point  x, 2.73 . Determine the value of x to
the nearest whole number.
7. Write as a single logarithm using the laws of logarithms.
b) log a B  log a C  log a D  log a E
a) log A  3log B  5log 5 C
8. Solve the following equations. Express answers as exact values.
a) log x 64  
2
3
b) log8 x 
b g b g
log b x  4g   log b x  2g  log 3
d) log6  x  3  1  log6  x  2 
c) log5 x  1  log5 x  3  1
e)
5
5
2
3
f) log3  4 x  8  log3  x  2   log3 9
5
g) 12 x1  5x
h) log9 x  log 4 64
9. Ethanol is a high octane, renewable fuel derived from crop sources, such as, corn or wheat.
Through the process of fermentation, yeast cells duplicate in a bioreactor and convert
carbohydrates into ethanol. In an experiment, researchers start with a yeast cell concentration
of 4.0 g/L in a bioreactor. Eight hours later, the yeast cell concentration is 12.8 g/L.
Algebraically determine the doubling time of the yeast cells, to the nearest tenth of an hour.
bg
bg
t
10.
The number of bacteria in a test tube is determined by the function N t  N o 2 d , where No
is the initial number of bacteria and d is the doubling time. A scientist began with 100
bacteria and estimates that there are 5000 bacteria immediately after 12 h. Determine the
doubling time, correct to the nearest tenth of an hour.
11.
Determine the age of a mammoth whose bones have lost one tenth of their original carbon-14
t
 1 h
mass using the function N  t   N 0   . The half-life of carbon-14 is 5800 years. Round
2
to the nearest whole year.
pg. 34
Math 30-1 Notes
Unit 6: Exponents and Logarithms
12.
The current population of the town of Winterlog is 3600. According to studies conducted
over the last few years, Winterlog’s population is growing at the rate of 1.6% per annum.
Assuming that the population will continue to grow at this rate, use the compound interest
formula to determine how many complete years it will take Winterlog to reach a population
of 4800.
13.
a) Compare the intensity of an earthquake of magnitude 8.5 on the Richter Scale to one with
I
I
a magnitude of 6 using the formula m2  m1  log or 10m2 m1  .
I0
I0
b) An earthquake was 4 times as strong as another which measured 8.3 on the Richter Scale.
What was the magnitude?
14.
a) The graph of y  log x is transformed to the graph of y  log  2 x  10  by a horizontal
stretch by a factor of p followed by a horizontal translation of q units. Determine the
value of p and q.
b) The graph of y  log5 x is translated 2 units down. A student writes the equation of the
transformed image in the form y  log5 kx . Determine the value of k, to the nearest
hundredth.
15. Write the following formulas using logarithms in terms of the variable t.
a) R  4ct
16.
b) V  1000 1.04 
t
The equation that defines the decibel level for any sound is
 I 
L  10log10   , where
 I0 
L = loudness in decibels
I = intensity of sound being measured
I0 = intensity of sound at the threshold of hearing
Given that normal conversation is 1 000 000 times as intense as I 0 , then determine the loudness of
normal conversation.
pg. 35
Math 30-1 Notes
Unit 6: Exponents and Logarithms
SOLUTIONS
5
2
1. a) x  1
b) x 
1
2. a) log3    4
 81 
b) log a z  x  y
3. a) r h  m
b) y  4 10 
4. a) 1
b)
5. a) 3x
b) 0.8
6. a) x  1
b) y 
c) f ( x)  3x :
f ( x)  log3 x :
domain:
34
3
a
c) log c    d
b
x
1
8
1
b2
domain:
d) x 
c) x  27
c) 25
d) 2
e) 3
c) 4y  x
d) 2M
e) 14
f)
x | x  R , range:  y | y  0, y  R
x | x  0, x  R , range:  y | y  R
d) x  20
 AC 
7. a) log  3 
 B 
8. a)
1
512
e) 5
 BC 
b) log a 

 DE 
b) 4
c) 4
f) no solution
g) x 
d) 3
 log12
log12  log 5
9. 4.8 hrs
10. d  2.1 h
11. 882 years
13. a) 316 times
b) 8.9
1
14. a) p  , q  5
2
 log R
15. a) t 
c log 4
12. 19 years
b) 0.04
b) t 
log V  3
log1.04
16. 60 decibels
pg. 36
h)
729
29
 4.83
6