log 5 (25*5^-3)

Question

log sub 5

how is the answer: -1

Answer 1

Question: log 5 (25*5^-3) : log sub 5 :  how is the answer: -1 ?

Let n = log_5[25*5^(-3)]

Take the expression inside the square brackets. We have 25*5^(-3)

and 5^(-3) = 1/(5^3) = 1/125

So 25*5^(-3) = 25*(1/125) = 1/5

and 1/5 = 5^(-1).

Our log expression then becomes log_5[5^(-1)] = n

Now, by definition, if log_a[b] = x, then b = x^a.

Since we have log_5[5^(-1)] = n, then, by definition, 5^(-1) = 5^n.

From which it follows: n = -1