Logarithmic Functions

The inverse of the equation

y = bx is x = by

Since there is no algebraic method for solving x = by for y in terms of x, the Logarithmic Function is used to allow y to be expressed in terms of x.

That’s right! Interchange x and y.

Sounds pretty easy so far. Let’s move on.

Let’s Take a Closer Look at Some Logs

A logarithm is really an exponent written in a different form.

The equation y = bx is an exponential function

Let’s break this down.

b is the basex is the exponenty is the value of bx

Now let’s bring in the logs.

Written in logarithmic form, the equation y = bx

would bex = log b aWe read this

asx is the logarithm of a with base b

Breaking Down Logs

Let’s look at a log piece by piece.

The equation x = log b

ais a logarithmic function

Let’s break this down.

b is the basex is the exponenta is the value of bx

Hey! I’ve seen this before.

It’s Sam Ting

as breaking down exponential functions.

That was easy

Comparing Logarithmic form and exponential

form

Exponential Form Logarithmic Form

y = bx

x = log b a

32 = 25 5 = log 2 32

512 = 83 3 = log 8 512

4 = log 3 81

3 = log 5 125

81 = 34

125 = 53

AsiDe

Facil

Logarithms with Variables

3 = log 4 a

In each equation, find the value of the variable

since 43 = 64, a = 64

x = log 6

36

since 62 = 36,

x = 2

3 = log b 125

since 53 = 125,b = 5

Hey, I can just use my calculator for this.

This looks a little harder. Maybe I should use a real calculator for this one.

That was easy

43 = a 6x = 36 b3 = 125

More Logarithms with Variables

In each equation, find the value of the variable

5 = log 8 a

since 85 = 32,768, a = 32,768

x = log 7 2,401

since 74 = 2,401,x = 4

3 = log b

6,859

since 193 = 6,859,b = 19

Hey, those are some pretty big numbers. I hope my calculator knows how to do this.

That was easy

85 = a 7x = 2,401 b3 = 6,859

Common Logs

Any logarithm with base 10 is a Common Log

When writing a common logarithm, the base is usually omitted.

So, 5 = log 10 100,000 and 5 = log 100,000 are Sam Ting.

Let’s compare Logarithmic Form and Exponential Form of some Common Logs.

Exponential FormLogarithmic Form

3 = log 1,000 1,000 = 103

1,000,000 = 1066 = log 1,000,000

10,000 = 104

4 = log 10,000

That was easy

Common Logs with Variables

In each equation, find the value of the variablex = log 100

10x = 100

since 102 =

100, then x = 2count the zeros

7 = log a

107 = a

since 107 = 10,000,000, then a = 10,000,000

write the proper number of zeros

Hey, I don’t even need a calculator for this!

That was easy

More Common Logs with Variables

Find the value of the variable to the nearest one hundredthx = log 1,345

10x = 1,345

Hey, there’s no zeros to count.

2.865 = log a

102.865 = a

That was easy

We could use the LOG key on our calculator.

LOG (1,345) = 3.13

What’s the proper number of zeros?We could use the 10x key on our calculator.102.865 = 732.82

Change of BaseHow can I get my calculator to evaluate logs in bases other than base 10? That’s easy, just use

the Change of Base Formula

logbx a

x = log 8 512 = 3

x = log 12 248,832 = 5

It’s time to push the easy button once again!

loglog

ab

log512

log8

248log ,832log12

More Change of Base

Let’s throw some decimals into the mix.

x = log 4 32

x = log 4.5 91.125 = 3

= 2.5

This stuff is too easy. Soon I’ll have to buy new batteries for my easy button.

x = log 8.125 1,986.597 = 3.625

That was easy

91log .125log4.5

log32

log4

1,98log 6.597log8.125