Chapter 8

EXAMPLE 6
Selecting strategies to simplify logarithmic
expressions
x 3 y 2
Use the properties of logarithms to express log a in terms of log a x, log a y,
Å w
and log a w.
Solution
x 3 y 2
log a
Å w
5 1 2 log a ax 3 y 2
w b
1
5 log a a x 3 y 2 2
w b
Express the square root using the
1
rational exponent of .
2
Use the power law of logarithms to
write an equivalent expression.
5 1 2 (log ax 3 y 2 2 log a w)
Express the logarithm of the quotient
of x 3 y 2 and w as a difference.
5 1 2 (log ax 3 1 log a y 2 2 log a w)
Express the logarithm of the product
of x 3 y 2 as a sum.
5 1 2 log ax 3 1 1 2 log ay 2 2 1 2 log aw
Expand using the distributive
property.
5 1 2 3 3 log ax 1 1 2 3 2 log ay 2 1 2 log aw
Use the power law of logarithms
again to write an equivalent
expression where appropriate.
5 3 2 log ax 1 log a y 2 1 2 log aw
Simplify.
In Summary
Key Ideas
• The laws of logarithms are directly related to the laws of exponents, since
logarithms are exponents.
• The laws of logarithms can be used to simplify logarithmic expressions if all the
logarithms have the same base.
Need to Know
• The laws of logarithms are as follows, where a . 0, x . 0, y . 0,
and a 2 1:
• product law of logarithms: log a xy 5 log a x 1 log a y
• quotient law of logarithms: log aQ x 5 log a x 2 log a y
y R
• power law of logarithms: log a x r 5 r log a x
474 8.4 Laws of Logarithms
NEL