Logarithmic Functions

The statement 8 = 2³ is in exponential form with base 2.

An equivalent way of writing 2³ = 8 is the logarithmic form:

We read log₂8 = 3 as, “the logarithm to the base 2 of 8 is 3.” Or, more briefly, “the log base 2 of 8 is 3.”

The statements 8 = 2³ and log₂8 = 3 are two ways of expressing the same relationship between 8 and the cube of 2.

If we rewrite 8 = 2³ in logarithmic form we have log₂8 = 3. So the logarithm, log₂8, is equivalent to 3, an exponent.

Definition — Logarithmic Function

A logarithmic function is a function that has the form: f(x) =  log_bx where b is a real number, b > 0, and b ≠ 1; x is a real number, x > 0.

The domain is all positive real numbers.

The range is all real numbers.

The inverse of an exponential function is a logarithmic function. This means that

• If f(x) = b^x then f^-1(x) = log_bx. Both functions have base b.

• The domain of one function is the range of the other.

• Their graphs are mirror images of each other about the line y = x.

Since they are inverses, we can write an exponential equation as a log equation and vice versa.

Note:

The inverse of f(x) = 2^x is f^-1(x) = log₂x.

The inverse of f(x) = log₂x is f^-1(x) = 2^x.

Definition — Exponential and Logarithmic Forms

Example 1

Rewrite in logarithmic form:

a. 5^x = 625

b. 3⁴ = x

c. b² = 100

Solution

The statements are given in exponential form, b^L = x.

Rewrite them in the equivalent logarithmic form, log_bx = L.

Example 2

Rewrite in exponential form:

a. log₆x = 2

b. log₄64 = x

c.

Solution

The statements are given in logarithmic form, log_bx = L.

Rewrite them in the equivalent exponential form, b^L = x.

Note:

Remember,