﻿ Properties of Logarithms

Properties of Logarithms

1. Properties of Logarithms and their proofs

 Property 1:   logbxy = ylogbx

Proof:

We have

logbxy = logb(blogb(x))y

=   logb(bylogb(x)) = ylogbx

 Property 2:   logb(xy) =  logbx +  logby

 Property 3:    logb(x/y) =  logbx -  logby

Exercise:

Prove properties 2 and 3.

2. Examples

Expand

Solution:

We have

ln(3x3)1/2 = 1/2 ln(3x3)                (Property 1)

= 1/2ln3 + 1/2lnx3                       (Property 2)

= 1/2ln3 + 3/2lnx.                        (Property 1)

Exercises:  Expand the following:

1. log[(x2(x - 4)5)/100]

2. log3(sqrt(x5/9))

Example:

Write the following with only one logarithm:

3log4x - 5log4(x2 + 1) + 2log4x2

Solution:

We use the properties:

log4x3 - log4(x2 + 1)5 + log4(x2)       (Property 1)

=    log4[x3/(x2 + 1)5] + log4(x4       (Property 3)

=    log4[x3x4/(x2 + 1)5                    (Property 2)

=    log4[x7/(x2 + 1)5]                        (A Property of Exponents)

Exercises:

Write the following with only one logarithm:

1. 2log3x - 2log3sqrt(x) + 5log31/x

2. logx - 2log(x - 1) + log(x + 1)

3. Application

The Rictor scale for earthquakes is as follows:  if I is the intensity of an earthquake and I0 is the intensity of the shaking without an earthquake, then the magnitude R of an earthquake is defined by

R = log[I/I0]

The Loma Prieta quake measured 7.1 on the Rictor scale and the Hokkaido quake measured 8.2.  How many times more intense was the Hokkaido quake?

Solution

Let

IL = The intensity of the Loma Prieta quake

and

IH = The intensity of the Hokkaido quake

We write

log(IH/IL)  = log(IH/I0 / IL/I0)

=  log(IH/I0) - log(IL/I0)

=  8.2 - 7.1  =  1.1

By exponentiating both sides with base ten, we get

IH/IL  = 101.1  =  12.6

We can conclude that the Hokkaido quake was more than 12 times more intense than the Loma Prieta quake.