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P ROOF OF THE P RODUCT R ULE Let M = b x and let N = b y. Then log b (M) = x and log b (N) = y. In addition, MN = b x+y by the properties of exponents. Taking the base b logarithm of both sides of our previous equation, we get log b (MN) = x + y. Substituting in log b (M) for x and log b (N) for y, we get log b (MN) = log b (M) + log b (N).
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THE PRODUCT RULE FOR LOGARITHMS
WHAT IS THE PRODUCT RULE FOR LOGARITHMS? Simply stated, the product rule for logarithms
is this:logb(xy) = logb(x) + logb(y)
This rule can prove very useful for simplifying logarithms.
PROOF OF THE PRODUCT RULE Let M = bx and let N = by. Then logb(M) = x and logb(N) = y. In addition, MN = bx+y by the properties of
exponents. Taking the base b logarithm of both sides of
our previous equation, we get logb(MN) = x + y.
Substituting in logb(M) for x and logb(N) for y, we get logb(MN) = logb(M) + logb(N).
EXAMPLE We can use the product rule to solve many
logarithm problems. For example, what is log10(125) + log10(8)? Using the product rule,
log10(125) + log10(8) = log10(125 * 8) log10(125 * 8) = log10(1000) = 3.
EXAMPLE What is log3(6) – log3(2)? Using the product rule, log3(6) = log3(3) +
log3(2). Thus, we have log3(3) + log3(2) – log3(2),
which is equal to log3(3), which, of course, is 1.
