5.3 Intro to Logarithms

2/27/2013

Definition of a Logarithmic Function

For y > 0 and b > 0, b ≠ 1,

logb y = x if and only if bx = y

Note: Logarithmic functions are the inverse of exponential functions

Example: log2 8 = 3 since 23 = 8Read as: “log base 2 of 8”

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form: x = logb y Exponential Form: bx = y

Exponent Exponent

Base Base

Basic Logarithmic Properties Involving One

• Logb b = __because 1 is the exponent to which b must be raised to obtain b. (b1 = b).

• Logb 1 = __because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

1

0

logb y = x if and only if bx = y

Popular Bases have special names

Base 10

log 10 x = log x is called a common logarithm

Base “e”

log e x = ln x is called the natural logarithm

or “natural log”

e and Natural Logarithme is the natural base and is also called “Euler’s

number”: an irrational number (like ) and is

approximately equal to 2.718281828...Real Life Use: Compounding Interest problemRemember the formula as n approaches +

The Natural logarithm of a number x (written as “ln (x)”) is the power to which e would have to be raised to equal x.

For example, ln(7.389...) is 2, because e2=7.389Note: and ln(x) are inverse functions.

Inverse properties

Since and ln(x) are inverse functions.

and

Since are inverse functions.

and Proof: Proof: Then

Example 1 Rewrite in Exponential Form

LOGARITHMIC FORM

a. log2 16 =

EXPONENTIAL FORM

4 24 = 16

b. log7 1 = 0 70 = 1

c. log5 5 = 1 51 = 5

d. log 0.01 = 2– = 0.0110 2–

e. log1/4 4 = 1– = 44

1 – 1

logb y = x is bx = y

Example 1 Rewrite in Exponential Form

LOGARITHMIC FORM EXPONENTIAL FORM

f. ln

log e x = ln x

𝑙𝑛𝑒2=log𝑒𝑒2=2 =

g. ln ln 𝑥=log𝑒𝑥=2 =

Example 2 Rewrite in Logarithmic Form Form

LOGARITHMIC FORMEXPONENTIAL FORM

log𝟕49=2a. =

logb y = x is bx = y

log .01=−2b. =

log𝟔1=0c. =

log1

10000=− 4d. =

Example 3 Evaluate Logarithmic Expressions

Evaluate the expression.

a. log4 64 4? = 64 What power of 4 gives 64?

43 = 64 Guess, check, and revise.

log4 64 = 3

logb y = x is bx = y

41/2 = 2 Guess, check, and revise.

log4 2 =2

1

4? = 2 What power of 4 gives 2?

b. log4 2

Example 3 Evaluate Logarithmic Expressions

= 93

1 –2

Guess, check, and revise.

= 93

1What power of gives 9?

?

3

1

log1/3 9 = 2–

c. log1/3 9

Since

d.

Example 4 Simplifying Exponential Functions

a. 7 log7 5

Since = 5

b. 2log2 √3

Since =

Example 4 Simplifying Exponential Functions

c. 𝑒ln 6

Since = 6

d. 𝑒ln 𝑥3

Since =

Homework WS 5.3 odd problems only