Logarithms

Inverse of an Exponential Function:

Find the inverse function of y = 2 ^x:

x = 2 ^y

There is no way to solve for y algebraically, so mathematicians have introduced a new word to solve this equation: logarithm

log₂x = y

This means the logarithm (or exponent) of x in base 2 is y.

Example: log₅25 = 2 because 5² = 25

Natural Logarithm:

ln x = k because e^k= x (e ~ 2.718)

ln always means a logarithm in base e

Common Logarithm:

log x = b because 10^b = x

log with no base indicated always means a logarithm in base 10

Graph of y = log x:

Graph y = log x in your calculator. It shows the following properties:

Domain: x > 0

Range: All Real Numbers

Asymptote: x = 0 (the y-axis)

It is a 1-1 function

It passes through (1,0)

Notice that properties of alogarithm graph interchange x and y in the properties for an exponential graph. This happens because they are inverse functions.

Problems