# Prove the product property of logarithms using the properties of exponents. `log_(b)m/n=log_(b)m-log_(b)n`

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### 1 Answer

Please see the attachment:

(a) 1. Set right hand side equal to x. We will show that x is the left hand side also.

2. Use the power property of logarithms.

3. Exponentiate both sides with base b.

4. Use a property of powers: a^(m+n)=a^m*a^n

5. Note that exponents and logarithms are inverse operations.

6. Take a logarithm of base b of both sides.

7. Exponents and logarithms are inverses.

8. Substitute for x from (1).

QED

(b) If you really wanted the product rule log(mn)=log(m)+log(n) you would use a similar procedure.

log(m)+log(n)=x

b^(log(m)+log(n))=b^x

b^(logm)*b^(logn)=b^x

mn=b^x

log(mn)=log(b^x)

log(mn)=x=log(m)+log(n) as required.

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