## 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

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

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,

For common logarithms, those with base 10, the base on the logarithm is often left out and written as

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.

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.

**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,

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.

**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

**Solution** Even though the input and output have variables, we may still reverse the roles and write logarithmic form as

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,

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. *e*^{0} = 1

**Solution** The logarithmic form for this exponential form is

ln(1) = 0

**Solution** In this exponential form, groups of variables play the role of input and output. The input in the exponential form is and the output is the fraction The corresponding natural logarithm is

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.

**Solution** This form converts to an exponential form with a base of 4. The input on the logarithmic form is 64 and the output is 3. The corresponding exponential form is

**Solution** Before identifying the input and output, isolate the logarithm. This is done by subtracting *b* from both sides to yield

In this form, the input to the logarithm is *x* and the output is *y* – *b*. The exponential form is

**Solution** The input on this common logarithm is the group and the output is *M*. Switching these roles gives the exponential form