Give examples of each?

Question

Explain the difference between a logarithms of a product and the product of logarithms and give examples of each.

Answer 1

logarithm of product is, for example, log(a*b) (assuming base 10). This can be separated into:
log(a*b) = log(a) + log(b).
Imagine the log expression to be an exponent expression, let's say, 10^(A) * 10^(B). Now this equals 10^(A+B), rules of basic algebra. So think of logs as the A and B of the expression.

Product of logs is different. Here, you're multiplying two exponents. This would be like saying:
10^(A*B)
You can't separate this in any way as a product of 10's.

Answer 2

Log of products --> Log (ab) = log(a)+log(b)

Product of logs --> log(a)*log(b)

let a = 2, b =3

log(2*3) = log(6) =log(2)+log(3) = 0.77815

log(2)*log(3) = 0.1436

Answer 3

logarithm of a product = log(xy) = log x + log y
product of logarithms = (log x) (log y) = log (x^(log y)) = log (y^(log x))

Suppose x = 10^a, y = 10^b, then
log (xy) = log (10^a.10^b) = a + b
(log x) (log y) = (log 10^a) (log 10^b) = ab.