expanding logarithms using the property rules

Pre-Calculus Tutorial

expanding logarithms using the property rules

Intro

Before attacking this problem, you should be familiar with these three rules:

“b” represents the base

1: Product Rule:
log b (XY) = log b X + log b Y

2: Quotient Rule:
log b (x/y) = log b X – log b Y

3: Power Rule:
log b M^p = p log b M

Sample Problem

log b (X^2 Y/Z^6)

2 log b Y + log b Z – 6 log b X


2 log b Y + log b X – 6 log b Z


6 log b X + log b Y – 2 log b Z


6 log b Z + log b Y – 2 log b X


2 log b X + log b Y – 6 log b Z


Solution

log b (X^2 Y/Z^6)

The first step is to use the quotient rule to eliminate the fraction
1: log b (X^2 Y/Z^6) => log b X^2 Y –log b Z^6

The next step is to use the product rule
2:log b X^2 Y –log b Z^6 => log b x^2 + log b Y – log b Z^6

The third and last step is to use the power rule
3: log b x^2 + log b Y – log b Z^6 => 2 log b X + log b Y – 6 log b Z



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