Properties of Logarithms

I.  Homework

II.  Properties of Logarithms and their proofs

Property 1:  log_bx^y = ylog_bx

Proof:   We have

log_bx^y = log_b(b^logb(x))^y =   log_b(b^ylogb(x)) = ylog_bx

Property 2:  log_b(xy) =  log_bx +  log_by

Property 3:   log_b(x/y) =  log_bx -  log_by

We will prove Properties 2 and 3 in groups.

III.  Examples

Expand ln(sqrt(3x³))

Solution:  We have ln(3x³)^1/2 = 1/2 ln(3x³) = 1/2ln3 + 1/2lnx³ = 1/2ln3 + 3/2lnx.

Exercises:  Expand the following:

A)  log[(x²(x - 4)⁵)/100]

B)  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)²

=    log₄[x³/(x² + 1)⁵] + log₄(x⁴)

=    log₄[x³x⁴/(x² + 1)⁵]

=    log₄[x⁷/(x² + 1)⁵]

Exercises:  Write the following with only one logarithm:

A)  2log₃x - 2log₃sqrt(x) + 5log₃1/x

B)  logx - 2log(x - 1) + log(x + 1)

IV.  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?