Question

-Explain the difference between log base b of m plus log base b of n and log base b (m+n). Are they equivalent?

-Explain the difference between log base b (mn) and ( log base b of m)(log base b of n). Are they equivalent?

Answer 1

Is log_b m + log_b = log_b(m+n)

no they are not, since
log_bm + log_bn = log_b(mn)

All we need is ONE counter-example to show that your statement is false.

Assume that
log_b m + log_b = log_b(m+n)

then it should be true for all values of b, m, and n within the domain of log definitions.

let's take some values for which we know the exact values
e.g.
log₂ 8 + log₂ 16
= 3 + 4
= 7
and log₂ (8+16)
= log₂ 24 ≠ 7

Use a similar argument and example for your second question.