MATH 11218
WEEK 4
CHAPTERS 8
LOGARITHMS AND EXPONENTIALS
Exponential Expressions:
In an exponential expression a x , a is the base and x is the exponent/power/index.
We will deal with exponential expressions of the form e kx
where e is the exponential constant 2.7183 (to 4dp) and k is a constant.
To simplify exponential expressions we use the index laws as shown in the notes for Week 1- Chapter 5.
Exponential Functions:
An exponential function is of the form y = a bx where a is the base, bx is the exponent and a, b are constants
with a > 0 .
The standard exponential function is y = e x
The graph of the function y = e x approaches the x axis as x → - ∞ , cuts the y axis at (0, 1) and approaches ∞
as x → ∞ (see graph on page 258)
Note:
ex 
1
ex
The graph of y = e –x is a reflection in the y-axis of the graph of y = e x
Applications of Exponential Functions:
If we have exponential growth, the rate of growth is proportional to the amount present at any time.
The formula for the amount present at any time t is N(t) = N0 e kt where N0 is the initial amount.
If we have exponential decay, the rate of growth is also proportional to the amount present at any time but the
formula is N(t) = N0 e - kt where N0 is the initial amount.
With Newton’s Law of Cooling, the rate of loss of temperature is proportional to the difference between the
temperature of the body at ant time, (t), and that of the surrounding environment, e.
The formula for the temperature at any time is
(t) = e + (0 - e) e - kt where 0 is the initial temperature of the body (ie at t = 0)
and k is the positive constant depending on the material of the body
Hyperbolic Functions:
Hyperbolic functions are defined as combinations of exponential functions
hyperbolic cosine
cosh x = ½ ( e x + e –x )
pronounced ‘cosh’
hyperbolic sine
sinh x = ½ ( e x - e –x )
‘shine’
hyperbolic tangent
hyperbolic secant
hyperbolic secant
hyperbolic cotangent
sinh x
e x  e x
 x
cosh x
e  e x
1
sech x 
cosh x
1
x0
cosech x 
sinh x
1
x0
coth x 
tanh x
tanh x 
‘than’
‘sheck’
‘cosheck’
‘coth’
From these hyperbolic functions, we can prove the identities
and
cosh x + sinh x = ex
cosh x - sinh x = e -x
These expressions are used to express exponential expressions as hyperbolic expressions and to prove
other hyperbolic identities.
Reference: Table 1.5on p266 and Example 1.9 on p264
To prove an identity, start with the more complicated side, manipulate and simplify it and prove that the
expression can be shown to equal the other (simpler) side.
Note: The equation for a catenary is
y = a cosh( x/b )
as shown on p265
Logarithms:
A logarithm is a power or index.
Definition:
If N = ax
then
x = log a N
ie x is the logarithm of N to base a.
Common logarithms are to base 10 and the base is usually not written. It is understood eg log 1000 = 3
Natural logarithms are to base e and are written as ln
Log Laws:
(ie
log e 5 = ln 5 )
log a ( xy)  log a x  log a y
x
log a    log a x  log a y
 y
log a x n  n log a x
x  a loga x
log b x
log a x 
log b a
(Change of base theorem)
log 1  0
ln ex = x
e ln y = y
To solve log equations, we use the definition to change to index equations, and visa versa.
Reference: Examples on pp279 - 282
Logarithmic Functions:
The logarithmic function y = logax is the inverse of the exponential function y = a x.
The logarithmic function y = ln x is the inverse of the exponential function y = e x.
The graph of the logarithmic function is a reflection in the line y = x of the graph of the exponential function.
Reference: Graph on page 277
Note: The function is only defined for x > 0
Applications of Logarithms:
If the level of intensity of a sound (in decibels) is I, relative to a reference level, I0 then the level of the sound
(or the power gain) in decibels is
x  10 log
Reference: pp 285 - 286
I
I0