7.3 LOGARITHMIC FUNCTIONS AS INVERSES

Part 1: Introduction to Logarithms

Logarithms

The inverse of the exponential function is the logarithmic function. By definition, y = bx is equivalent to logby =

x. Logarithms exist only for positive real numbers.

Logarithms

The definition of a logarithm can be used to write exponential functions in logarithmic form: y = bx is equivalent to logby = x

Example:

Write each equation in logarithmic form100 = 102

y = bx is equivalent to logby = x

Example:

Write each equation in logarithmic form81 = 34

y = bx is equivalent to logby = x

Logarithms

To write a logarithmic function in exponential form, use the definition:

If y = bx is equivalent to logby = x,

then logby = x is equivalent to y = bx

Example

Write each equation in exponential form.log2128 = 7

logby = x is equivalent to y = bx

Example

Write each equation in exponential form.log716,807 = 5

logby = x is equivalent to y = bx

Logarithms

The exponential form of a logarithm can be used to evaluate a logarithm.1. Write a logarithmic equation (set the

log = x)2. Use the definition to write the

logarithm in exponential form3. Write each side of the equation using

the same base4. Set the exponents equal to each

other5. Solve

Example:

Evaluate each logarithmlog5125

Example:

Evaluate each logarithmlog832

Example:

Evaluate each logarithm

Common Logarithm

The common logarithm is a logarithm with base 10log10

The common logarithm can be written without a base, because it is understood to be 10log10x = log x The “log” key on your calculator is the common

logarithm

Homework

P456 #12 – 35