Chapter 8

log a (mn) 5 x 1 y
m 5 a x so log a m 5 x
n 5 a y so log a n 5 y
Write the powers involving m and n
in logarithmic form. Substitute the
logarithmic expressions into the
equation log a (mn) 5 x 1 y.
log a (mn) 5 log a m 1 log a n
The logarithm of a product is equal
to the sum of the logarithms of the
factors.
EXAMPLE 2
Connecting the quotient laws
Determine an equivalent expression for log a Q m , where a, m, and n are positive
n R
numbers and a 2 1.
Solution
Since a, m, and n are all positive, m
Let m 5 a x and n 5 a y .
and n can be expressed as powers
of a.
m
n 5 a x
Substitute the expression for m and
a 5 a x2y m
n into the quotient . Simplify using
y n
the quotient law for exponents.
log a a m n b 5 log a(a x2y )
log a a m n b 5 x 2 y
m 5 a x so log a m 5 x
n 5 a y so log a n 5 y
log a a m n b 5 log am 2 log a n
These expressions must be equal
m
since , as shown above.
n 5 a x2y
On the right side of this equation,
the exponent that must be applied
to a to get a x2y is x 2 y.
Write the powers involving m and n
in logarithmic form. Substitute the
logarithmic expressions into the
equation log a Q m .
n R 5 x 2 y
The logarithm of a quotient is equal
to the logarithm of the dividend
minus the logarithm of the divisor.
470 8.4 Laws of Logarithms
NEL