Logarithmic Functions

1

Inverse Functions An inverse function “undoes” whatever the function does The inverse of the exponential function is the logarithmic function That is if

bx

f x

y then logb y

f x

bx

logb x x

2

Inverse Function Function

x

y

bx

y

Inverse Function

x

y

logb y

x

3

1

Logarithmic Function For x

0, b

0, and b

to the base b is y

1,the logarithmic function

log b x which is definded by x

bx

This function is the inverse function of the exponential bx

function y

4

LOG IS ANOTHER WAY TO WRITE AN EXPONENT by

y logb x

x

b

the base

y

the exponent

x

what the exponential

expression is set equal to

Example

Write the inverse of x

y

5

3y

log3 x

6

2

Logax and ax are Inverse Functions f x

1

f

ax y

a

x

loga x y

x

x

y=loga x

We know f f So, aloga x

1

x

1

x and f

x and loga a x

f x

x

x

7

Logarithmic Functions • The range is R • The domain is the set of strictly positive real numbers • The function is continuous in its domain • The function is increasing if a > 1 and decreasing if 0