Question 1: How do you convert between the exponential and logarithmic forms of an
equation?
Exponential and logarithm functions are inverses of each other. In the simple terms, this
means that an expression in exponential form may be converted to logarithm form by
switching the inputs and outputs. Let’s start with a concrete example. The exponential
function
y  10 x
takes the variable x as its input and outputs the variable y. For an input of x  2 we get
an output of y  100 since
100  102
On a logarithm of base 10 (called a common logarithm), these roles are reversed. The
common logarithm must take in y  100 and output x  2 ,
2  log10 100 
For common logarithms, those with base 10, the base on the logarithm is often left out
and written as
2  log 100 
This means that whenever you see a logarithm without a base, it is assumed to have a
base of 10. Let’s compare these forms side by side.
Exponential Form
Logarithmic Form
100  102
2  log10 100 
output
input
output
input
2
The base on the exponential form is below the 2. On the logarithm form, the base is just
after and slightly below the word log. The exponential form take in 2 and outputs 100.
The logarithm form does exactly the opposite. It takes in 100 and outputs 2. The
numbers are the same, but they role they play is reversed.
For all bases b  0 ,
y  b x means that x  log b  y 
Example 1
Convert to Logarithmic Form
Convert each exponential form below to its equivalent logarithmic form.
a.
103  1000
Solution For this exponential form, the base is 10 so it will convert to a
logarithm base 10 or common logarithm. The input is 3 and the output
is 1000. When this exponential is converted to a logarithm, the input will
be 1000 and the output will be 3. This gives the logarithmic form,
3  log 1000 
Note that the original exponential form has the exponent on the left side
instead of the right. Where it appears is irrelevant. They key is to
recognize that the exponent is the input and the other side of the form is
the output.
b.
32  25
Solution This exponential form will convert to a logarithm with a base of
2. Since the input is 5 and the output is 32, the logarithmic form is
log 2  32   5
3
c.
5t  z
Solution Even though the input and output have variables, we may still
reverse the roles and write logarithmic form as
log 5  z   t The base e occurs frequently in business and finance. Like the constant π, the constant
e represents an irrational number whose value is approximately 2.718. Because this is
an approximation, you’ll often see the approximately equal sign,
, when using this
base. Most scientific and graphing calculators have an e button to help you evaluate
exponentials with a base of e.
Exponential forms with a base of e convert to logarithms with a base of e. For instance,
Exponential Form
Logarithmic Form
7.389  e 2
2  log e  7.389 
output
input
output
input
Logarithms with a base of e are also called natural logarithms. They are often
abbreviated by writing 2  ln  7.389  instead of 2  log e  7.389  .
Example 2
Convert to Logarithmic Form
Convert each exponential form with a base of e to logarithmic form.
a.
e0  1
Solution The logarithmic form for this exponential form is
4
ln 1  0
b.
A
 ert
P
Solution In this exponential form, groups of variables play the role of
input and output. The input in the exponential form is rt and the output
is the fraction
A
. The corresponding natural logarithm is
P
 A
rt  ln  
P
Logarithmic forms may also be converted to exponential form. As with exponential form,
they key is to identify the input and output and switch those roles.
Example 3
Convert to Exponential Form
Convert each logarithmic form to exponential form.
a.
log 4  64   3
Solution This form converts to an expopnential form with a base of 4.
The input on the logarithmic form is 64 and the output is 3. The
corresponding exponential form is
43  64
b.
y  b  ln  x 
Solution Before identifying the input and output, isolate the logarithm.
This is done by subtracting b from both sides to yield
y  b  ln  x 
5
In this form, the input to the logarithm is x and the output is y  b . The
exponential form is
e y b  x
c.
 A
M  log  
 A0 
Solution The input on this common logarithm is the group
A
and the
A0
output is M. Switching these roles gives the exponential form
10M 
A
A0
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