MA112
Section 106
Dr. Byrne
Fall 2009
The logarithm function is the inverse of the exponential function.
x  by 
y  log b x
(for x, b > 0 and b  1)
We say that y is the “log of x, base b”.
Worksheet 5.3: Calculating Logarithms
Calculate the following logarithms without using a calculator. Notice the inverse relationship of the
logarithm with the exponential function.
9  32
16  2 4
7  71
43  43
log 3 9  ______
log 2 16  ______
log 7 7  ______
log 4 4 3  _____
Answers: ‘how can we
get 9 by raising 3 to a
power’?
Answers: ‘how can we
get 16 by raising 2 to a
power’?
Answers: ‘how can we
get 7 by raising 7 to a
power’?
log 4 64  _____
log 16 4  _____
log 10 0.001  _____
Answers: ‘how can we
get 4
Answers: ‘how can we
get 43 by raising 4 to a
power’?
log 2
1
 _____43
8
A
Two most common logarithms.
log e x : ln x
log x : log 10 x
log 0.01  _____
Using your calculator.
You may need the
change of base formula:
log b M 
log a M
log a b
ln e  _____
log 4 32  _____ log 2 e  _____
ln e 2  _____
log 9 27  ____
Worksheet 5.3: The Natural Logarithm is the inverse of ex
The natural logarithm function is the logarithm with base e.
ln x  log e x
This means:
log e x  y
Suppose we have:
ln x  y
So we know:
y
(The RHS answers: ‘how can we
get x by raising e to a power?’)
The natural logarithm and the natural exponent are inverses.
We have:
ln e  x
x
and
e
ln x
 x.
e ln 5  _____
Sketch the graph of the function and its inverse.
y  ln x
y  ex
x
y
y  log 2 x
x
x
y
y  2x
y
x
y
ln e
xe
1
3
 _____
Worksheet 5.3: Converting Between Exponential and Logarithmic Equations
To convert an exponential equation to a logarithmic equation, take the logarithm of both sides. The
base of the logarithm will be the base of exponent.
93
2
16  2 4
7  71
43  43
Notice that this derives the same expressions that you find on the first page of this worksheet.
Converting an exponential equation to a logarithm helps us solve the equation when the variable is
in the exponent.
10 x  1000
8x  2
10 x  2
e 2  7.3891
To convert a logarithmic equation to an exponential equation, make each side of an equation the
exponent of a base b, where b is the base of the logarithm.
log 0.01  2
t  log 4 7
log 5 x  1
log x  0.845