Explain the difference between a logarithm of a product and the product of logarithms and give examples of eac

Answer 1

Imagine this...
log (a*b) = log a + log b

So, log (a^b) = log (a*a*a*a*a*... for a b number of times )...
therefore, given the first rule...
= log a + log a + log a + log a + .... for a b number of times)

= b * log a

Answer 2

A log of a product would be log(ab), which can be expressed as log(a) + log(b).

A product of logs would be [log(a)]·[log(b)], which might simplify, but rarely ever does.

Examples:
log[base10](20)
= log[base10](2·10) --note log of product...
= log[base10](2) + log[base10](10)
= 0.30103 + 1
= 1.30103

log[base10](5) · log[base10](2) --note product of logs...
= 0.69897 · 0.30103
= 0.21041

In general, logs of products work out more nicely that products of logs.

Answer 3

First consider Log(a*b) = log(a) + log(b) Let a = 5 and b = 7 just for kicks.

Then Log(a*b) = log(35) = 1.544 = log(5) + log(7) = 0.699+0.845 =1.544

Now consider log(a)*log(b). Can't really simplify this any more so let's plug in numbers:

log(5)*log(7) = 0.699*0.845 = 0.591