What do these pairs have in common?

sin and sin-1 + and -

÷ and •

cos and cos-1

tan and tan-1

2and

inverses

HW Check

LOGARITHMIC FUNCTIONS

Inverses undo each other!

So what undoes x when it’s an exponent?

y = a b∙ x Get me down!

Introducing… Logarithms!

A logarithm is defined as follows:

If y = bx, then logb y = x

x is now safely on the ground!

Example 1: Write 25 = 5x in logarithmic form.

Example 2: Write ⅛ = (½)x in logarithmic form.

Okay, we can get x down from the exponent, but what do we do with an expression like this?

log10 100 = x Your calculator automatically uses

log10 when we press the LOG function the

calculator. Try it!

How does this help us?

Change of Base Formula

What if we have an expression that doesn’t have a base of 10?

log9 81 = xChange of Base

Formula

log9 81 = 9log

81log

10

10

Example 3: Write 98 = 7x in logarithmic form. Then solve

Example 4: Write 42 = 9x+2 +7 in logarithmic form. Then solve.

Example 5: Write 56 = 5x-9 – 4 in logarithmic form. Then solve.

SOLVING LOGARITHMIC

EQUATIONS

So what undoes x when it’s an exponent?

y = a b∙ x

Yesterday, we learned how to solve for a variable when it is an

exponent.

Get me down!

What do we do if we have a variable trapped in a log?

So what undoes x when it’s an exponent?

log4 x = 78 Get me out

of here!

Rewrite it as a exponential function!

If logb y = x , then y = bx

No longer trapped inside the log!

Example 1: Solve log5 x = 2

Example 2: Solve 3log8 x = 3

Example 3: Solve log5 (x-2) = 8

Example 4: Solve log5 (x-2)+4 = 3

Example 5: Solve 2log5 (x+2) - 5= 3