exponential & logarithmic functions review
Math 30-1: Exponential & Logarithmic Functions – Unit 5
Review of the Exponent Laws
3  3  3  3  34
3 is the base, 4 is the exponent and 34 is a power.
Laws
am
1
 m
a
a
 a mn
n
a
a
 
b
m
a 
a
a n  n am
 ab 
 a mb m
a m  a n  a mn
m
m n
m
b
 
a
Laws of Logarithms
loga  M  N   loga M  loga N
m
m
mn
M 
log a    log a M  log a N
N
log a M n  n log a M
Change of Base Formula:
log a c
log b c 
log a b
Exponential Function

A function of the form y  c x , where c is a constant (c > 0) and x is a variable.
y  c x 
 y  a (c ) b ( x  h )  k
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Math 30-1
exponential & logarithmic functions review
Example
1 1x
Transform the graph of y  3x to sketch the graph of y   (3) 5  5 . Describe the
2
effects on the domain, range, equation of the horizontal asymptote, and intercepts.y  3x
Logarithms
Example
Evaluate.
a) log 7 49
b) log 6 1
Example
Change to logarithmic form.
x
a) y  3  5 
b) y  4 x  9
Collection of Useful Results

log a a  1

log a a n  n

log a 1  0


a loga n  n
We can’t take the log
of 0 or of a negative
number.
Exponential Function

A function of the form y  logb x , where b is a constant (b > 0)
and x is a variable.
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Math 30-1
exponential & logarithmic functions review
Example
a) Use transformations to sketch the graph of the function y  2log3   x  1 .
b) Identify the following characteristics of the graph of the
function.
i) The equation of the asymptote
y  log3 x
ii) The domain and range
iii) The y-intercept, if it exists
iv) The x-intercept, if it exists
Example
Write each expression in terms of individual logarithms of x, y, and z.
a) log5 xy
b) log 6
x5 y
z
Laws of Logarithms
loga  M  N   loga M  loga N
M 
log a    log a M  log a N
N
log a M n  n log a M
Example
Write as a single logarithm
a) log3 9  log3 18
b) 3log 2 4  2log 2 8  7log 2 2
Log Applications
• Earthquakes
• Sound Intensity
• pH Scale
Example
a) One earthquake measured 9.5 on the Richter scale and the second earthquake
measured 7.6. How much more intense was the first earthquake than the second.
To find
intensities
compare
earthquake points
b) A major earthquake of magnitude 7.5 is 375 times as intense as a minor
not Richter scales
earthquake. Find the magnitude, to the nearest tenth, of the minor earthquake.
c) A whisper (30 dB) is _____ times as loud as a conversation (60 dB).
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Math 30-1
exponential & logarithmic functions review
Solving Exponential & Logarithmic Functions
Examples
1
a) 27 x  2  x 3
81
b) 3  4 x1  15
d) log4  x  2  log 4  x  4   2
c) log 3  log x  log 2
RF 7 Demonstrate an understanding of logarithms.
RF 7.1
Explain the relationship between logarithms and exponents.
1.
The graph of y  3x is reflected in the line y  x . Determine the equation of the
transformed graph.
(RF 7.1)
Different Inverse Notations
- Inverse of a Relation
- y  f ( x) 
 x  f ( y)
For Inverse’s
- Reflection over the line y  x
switch x’s and
y’s
- y  f 1 ( x)
-
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Math 30-1
exponential & logarithmic functions review
2.
Given y  4 and its inverse x  4  y  log 4 x .
a) Complete the table of values for each
y  4x
y
x
-2
-1
0
1
2
x
y
(RF 7.1)
y  log 4 x
x
y
c) If f ( x)  log 4 x, sketch
g ( x)  2 f x  3
b) Sketch y  log 4 x .
RF 7.2 Express a logarithmic expression as an exponential expression and vice versa.
3.
Determine the exponential form of the equation m log p n  q .
(RF 7.2)
4.
5.
Express each in logarithmic form.
a) 5 x  25
(RF 7.2)
b) y  3  2
Express each in exponential form
a) log 4 a  2
x
(RF 7.2)
b) log m ( x  2)  5
RF 7.4 Estimate the value of a logarithm, using benchmarks, and explain the reasoning
6.
Rank these logarithms in order from least to greatest:
(RF 7.4)
log 4 62 , log6 36 , log3 10 , log5 20
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Math 30-1
exponential & logarithmic functions review
RF 8 Demonstrate an understanding of the product, quotient and power laws of
logarithms.
RF 8.3
Determine, using the laws of logarithms, an equivalent expression for a
logarithmic expression.
1
7.
The value of log 5 625  3log 7 49  log 2  log b b  log b 1 is ___________ (RF 8.3)
16
8.
Simplify the expression  3log x  3log x  to be one power.
9.
Write 2 log x 
10.
Given that log3 a  6 and log3 b  5 , determine the value of log 3  9ab 2  . (RF 8.3)
11.
Identify where the student made an error and correct the solution.
log z
 3log y as a single logarithm.
2
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(RF 8.3)
(RF 8.3)
(RF 8.3)
Math 30-1
exponential & logarithmic functions review
12.
Simplify and then evaluate each expression, round to four decimal places if
necessary.
(RF 8.3)
a) 3 log 6 2  2 log 6 9  log 6 3
b) log 4 1  log8 1
c) 3log3 6
2
d) log6 0
e) log 2  4
f) log 2 8  log 2 32
13.
Express log 2 200 in terms of x given x  log 2 5
(RF 8.3)
14.
Given 3  log x 8 , evaluate log x 32 .
(RF 8.3)
15. Using the equation log 27 x  y , what would the expression log 9 x equal? (RF 8.3)
16.
1
If log3 5  x , express 2log3 45  log3 225 in terms of x.
2
-7-
(RF 8.3)
Math 30-1
exponential & logarithmic functions review
RF 9 Graph and analyze exponential and logarithmic functions.
RF 9.2
Identify the characteristics of the graph of an exponential function of the
form y  a x , a > 0, including the domain, range, horizontal asymptote and
intercepts, and explain the significance of the horizontal asymptote.
x
1
17. Given that f  x     ,
3
a) Determine the domain of y  f  x  .
(RF9.2)
b) Determine the range of y  f  x  .
c) Determine any asymptotes in the graph of y  f  x  .
d) Determine the intercepts in the graph of y  f  x  .
e) Is y  f  x  an increasing or decreasing function? Explain how you know.
18. Determine the y-intercept on the graph of f  x   a x1  b .
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(RF9.2)
Math 30-1
exponential & logarithmic functions review
RF 9.1 Sketch, with or without technology, a graph of an exponential function of the
form y  a x , a > 0.
RF 9.3 Sketch the graph of an exponential function by applying a set of transformations
to the graph of y  a x , a > 0, and state the characteristics of the graph.
19. Given that f  x   2x :
a) Sketch the graph of f  x   2x .
(RF9.1)
b) Describe the transformations you would need to apply to the graph of y  2 x to
1
x 5
sketch the graph of y   2   1.
2
c) Sketch the graph of y 
1
x 5
 2   1.
2
(RF9.3)
(RF9.3)
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Math 30-1
exponential & logarithmic functions review
RF 9.6 Sketch the graph of a logarithmic function by applying a set of transformations
to the graph of y  logb x , b > 1, and state the characteristics of the graph.
20.
Given y  log 2 ( x  4)
(RF 9.6)
a) Determine the following characteristics
Domain:
Range:
Equation of the asymptote:
Coordinates of the x-intercept:
b) If f ( x)  log 2 x, what is the x-intercept of hx   f 13 x  ?
21.
Determine the equation of the asymptote for the graph of y  logb  x  3  2 ,
where b  0 and b  1.
(RF 9.6)
22.
Complete the sentence: The graph of y  log x is transformed into the graph of
y  4  log  x  5 by a translation of 5 units _______ and 4 units _______.
(RF 9.6)
23.
24.
Determine the domain of the graph of y  logb  3x  12 , where 0  b  1 . (RF 9.6)
The point (1, 0) is on the graph of y  log3 x . Determine the coordinates of the image
point after the graph has been translated according to the equation y  log 3 2 x  4 .
(RF 9.6)
25.
Describe in words the transformations required to transform the graph of
1

y  log5 x to the graph of y   log 5  x  6   3 .
(RF 9.6)
2

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Math 30-1
exponential & logarithmic functions review
RF 10 Solve problems that involve exponential and logarithmic equations.
RF 10.1 Determine the solution of an exponential equation in which the bases are
powers of one another.
26. Algebraically solve the equation 83x4  4 x9 .
If possible get a
common base
on both sides
27. Solve algebraically.
a) 3  3x  81
c) 9 log5 x  81
(RF 10.1)
b) 8
3 x 2
4
4 x 1
d) 52 x  110
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Math 30-1
(RF10.1)
exponential & logarithmic functions review
RF 10.2
Determine the solution of an exponential equation in which the bases are not
powers of one another, using a variety of strategies.
28. Graphically solve the equation 2 x1  3 2 x3 .
(Round your answer to 2 decimal places & sketch the graph on calculator)
29. Graphically solve the equation
(RF10.2)
 x 8 
3
 1  x 5
1000  
   3  .
4
 10 
(Round your answer to 2 decimal places.)
 2 x 1
30. Solve the equation 3
1
 
5
(RF10.2)
 x  3
algebraically. Round to the nearest hundredth, if
necessary.
(RF10.2)
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Math 30-1
exponential & logarithmic functions review
RF 10.3 Determine the solution of a logarithmic equation, and verify the solution.
31.
Solve algebraically log7  x  1  log7  x  5  1 and state any extraneous roots. (RF 10.3)
32.
For each of the following, solve for x. State any extraneous roots.
1
a) log 2 x  log 2 ( x  2)  3
b) log x  log 6  log
2
c) 2  log 9 x   log 9 x 7  4  0
2
RF 10.5
(RF 10.3)
d) 52 x 1  3x  2
Solve a problem that involves exponential growth or decay.
t
p
33. A radiactive isotope has a half-life of 7 years. Use the formula y  ab to determine
how much of the isotope must initially be present to decay to 60 grams in 14 years.
(RF 10.5)
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Math 30-1
exponential & logarithmic functions review
34. The population of a particular town on July 1, 2011 was 20000. If the population
decreases at an average annual rate of 1.4%, how long will it take for the population
to reach 15300?
(RF 10.5)
35. A bacteria culture starts with 5000 bacteria. After 6 hours, the estimated count is
80,000. What is the doubling period for this bacteria culture?
(RF 10.5)
36. The population of a city is increasing at a constant rate of 5% per year. The city’s
present population is 200 000. Determine the minimum number of years it will take
for the population to exceed 500 000.
(RF 10.5)
RF 10.6
Solve a problem that involves the application of exponential equations to
loans, mortgages and investments.
37. You invest $500 at 4% compounded annually.
(RF10.6)
a) Write an exponential function to model the value of your investment after n years.
Try Using:
y  ab x
b) Sketch a graph of your investment for the first twenty years.
c) Determine when your
investment will be worth
$1000.
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Math 30-1
exponential & logarithmic functions review
RF 10.7 Solve a problem that involves logarithmic scales, such as the Richter scale and
the pH scale.
38. Earthquake intensity is given by I  I 0  10 M , where I 0 is the reference intensity and
M is the magnitude. An earthquake measuring 5.3 on the Richter scale is 125 times
more intense than a second earthquake. Determine, to the nearest tenth, the Richter
scale measure of the second earthquake.
(RF 10.7)
39. The pH of lemon juice is 2.3 while the pH of milk is 6.6. Determine the concentration
of hydrogen ions in each substance using the formula pH   log  H   . How many
times more acidic is lemon juice than milk?
(RF 10.7)
40. The concentration of hydrogen ions in apple juice can be written as  H    3.2 104 .
Use the formula pH   log  H   to determine the pH of apple juice.
(RF 10.7)
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Math 30-1
exponential & logarithmic functions review
41. In 1812, an earthquake of magnitude 7.9 shook New Madrid, Missouri. How many
times more intense was the New Madrid earthquake compared to the 2001 earthquake
 I 
of magnitude 3.2 in Charlottesville, Virginia? m  log  
 Io 
(RF 10.7)
42. Scientists have found that the sensation of loudness can be described using a
logarithmic scale. The intensity level, in decibels, of a sound is defined as
I
, where I is the intensity of the sound, in watts per square metre and Io
LdB  10 log
I0
is 10-12 W/m2, corresponding to the faintest sound that can be heard by a person of
normal hearing. What is the measure of the loudness, in decibels of a hair dryer that
has an intensity of 10-5?
(RF 10.7)
43. One brand of ear plugs claims to block the sound of snoring as loud as 22 dB. A
second brand claims to block snoring that is eight times as intense. If the claims are
I
true, for how many more decibels is the second brand effective? LdB  10 log
I0
(RF 10.7)
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Math 30-1
exponential & logarithmic functions review
Answers
1. y  log3 x
2.
a)
x
2. c)
-2
-1
0
1
2
q
3. p m  n or
6.
p q  nm
14. 5
12. a) log 6 216  3 b) 0
15. y log9 27
x
1
16
1
4
1
4
16
1
16
1
4
1
4
16
4. a) log5 x  25 b) log 2
log5 20 log6 36 log3 10 log 4 62
Step 3
y
7. 7
8. 3log x
2
-2
-1
0
1
2
5. a) 42  a b) m5  x  2
9. log
x2 y3
z
10. 18
c) 36 d) No Solution e) No Solution f) -2
11. Error is
13. 2 x  3
17. a) D:  x x  R b) R:  y y  0, y  R c) H.A @
16. 3+x
y  0 d) no x-ints, y-int:  0,1 e) exponential decay
stretch by a factor of
y
x
3
y
18. a+b
19. Vertical
1
, horizontal translation 5 units left, & vertical translation 1 unit
2
down
20. a) D:  x x  4, x  R R:  y y  R V.A. @ x  4 , x-int=5 b) 3
x3
22. Left, up
23. x  4
24.  2.5,0 
21. V.A. @
25. Reflection about the x-axis, horizontal
stretch by a factor of 2, horizontal translation of 12 units left, & vertical translation 3 units down
26.
30
7
27. a) 3 b) 8 c) 25 d)
-2 is extraneous
33. 240g
38. 3.2
log 5 110
 1.46 28. 2.65
2
32. a) No Solution b) x 
34. 19 years
39. 19953
35. 1.5 hours
40. 3.5
29. 12.27
30. 0.98
31. x  6 ,
1
1
c) x  , 6561 d) x  log 25 45  1.80
12
3
3
36. 19 years 37. a) y  500(1.04)n c) 18 years
41. 50119
42. 70 43. 31 dB
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Math 30-1