Product Rule for Logarithms

Pre-Calculus Tutorial

Product Rule for Logarithms


Let’s show that log_a(c*d)=log_a(c) + log_a(d)

1. Let log_a(c)=x …. We can name it anything we like
2. Let log_a(d)=y …. Again we can name it anything we like

3. a^x=c and a^y=d … Equivalent exponential forms of the statements in steps 1 and 2.

4. a^x•a^y=c*d .. Reason: If A=B & C=D, then AC=BD.

5. a^{x+y}=c*d ..Reason: Property of exponents: AQ•AS=AQ+S.

6. log_a^{x+y}=(x+y)=log_a(c*d) .Reason: Taking log (base a) of both sides.

7. x+y=log_a(c*d) Reason: same expression as step 6

8. log_a(c) + log_a(d) = log_a(c*d). Substituting from steps 1 and 2 into step 7

Sample Problem

show that

log_2(8) + log_2(16) =  log_2(128)


log_2(8)=3 …. 2*2*2 = 8
log_2(16)=4 … 2*2*2 *2=16

log_2(8) + log_2(16) = 3 + 4 = 7 = log_2(128) ….. 2^7=128

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