LogarithmsEssential Question – How is a log function

related to an exponential function?You use log functions to solve exponential problems;

they are inverses of each other.

Math 3 Keeper 27

When will I use this?

• Human memory

• Intensity of sound (decibels)

• Finance

• Richter scale

Evaluating Log Expressions

• We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6?• Because 22<6<23 you would expect

the answer to be between 2 & 3.• To answer this question exactly,

mathematicians defined logarithms.

• Logarithms are the INVERSE of exponential functions.

Definition of Logarithm to base b

Let b & x be positive numbers, b ≠ 1.

logby = x iff bx = y

• This expression is read “log base b of y”

• The definition tell you that the equations logby = x and bx = y are equivalent.

Log form Exp. form

a) log232= 5

b) log51 = 0

c)log101 = 1

d) Log1/2 2 = -1

25 = 3250 = 1101 = 1(1/2)-1 = 2

Example 1: Rewrite the equation in exponential form

Log form Exp. form

e) log39= 2

f) log81 = 0

g)log5(/25)=-2

32 = 980 = 15-2 = 1/25

YOUR TURN!!

Rewriting forms:

To evaluate log3 9 = x ask yourself…

“3 to what power is 9?”

3x = 9 → 32 = 9 so…… log39 = 2

Example 2: Evaluate the expression without a calculator

a)log381

b) log50.04

c) log5125

3x = 81

5x = 0.04

5x = 125

4

-2

3

YOUR TURN!!

d) log4256

e) log464

f) log1/4256

g) log2(1/32)

4x = 2564x = 64(1/4)x = 2562x = (1/32)

4

3-4

-5

You should learn the following general forms!!!

•Log b 1 = 0 because b0 = 1

•Log b b = 1 because b1 = b

•Log b bx = x because bx = bx

Natural logarithms

log e x = ln x

• The natural log is the inverse of the natural base, e.

• ln means log base e

Common logarithms

log 10 x = log x

• Understood base 10 if nothing is there.

Common logs and natural logs with a calculator

log10 button

ln button

Example 3: Use a calculator to evaluate the expression. Round

answer to 3 decimal places.

a) log 5

b) ln 0.1

c) log 7

d) ln 0.25

0.6989= 0.700

-2.303

0.845

-1.386

g(x) = log b x is the inverse of the exponential function f(x) = bx

f(g(x)) = blogbx = x

g(f(x)) = logbbx = x

*Exponential and log functions are inverses and “undo” each other

INVERSE PROPERTIES

a) 10log2 = b) log39x =c) 10logx =

d) log5125x =

2log3(32)x =log332x=2xx

3x

Example 4: Using inverses→ Simplify the expression.

log5(53)x =log553x =

Finding Inverses

Find the inverse of y = log3x

• By definition of logarithm, the inverse is

y=3x

OR write it in exponential form and switch

the x & y! 3y = x → 3x = y

Example 5: Find the inverse of...

a) y = ln (x +1) X = ln (y + 1) Switch the x & y

x = loge(y + 1) Write in log form

ex = y + 1 Write in exp form

ex – 1 = y Solve for y

y = ex – 1 Final Answer

Example 5: Find the inverse of...

b) y = log8x 8y = x Switch x & y

8x = y Solve for y

y = 8x Final Answer

Example 5: Find the inverse of...

c) y = ln (x - 3)

d) y = log2/5x

e) Y = ln (x–10)

y = ex + 3

y = (2/5)x

y = ex + 10