I don't remember how to do this. Could someone please help? (Log base A of B)(Log base B of A) = Log base C?

using change of base formula,

(log[a] b) (log[b] a) =

(log b)/(log a) • (log a)/(log b) = .............. and here log is to any base at all

1 =

log[c] c .......... where c is any base at all (greater than 1, of course)

Suppose x = log [A] B , then

A^x = B. Take log base B of both sides:

x log[B] A = log[B] B = 1. Now replace x by what it equal (top line):

(log [A] B)(log[B] A) = 1.

So your question boils down to: 1 = log[C] (?). And the answer must be ? = C.

Note: I put bases in brackets.

Certainly.

Recall that, by the Change of base formula:

log_a(b) = log_c(b)/log_c(a) for any a, b, c in (1, infinity)

LHS = [log_A(B)][log_B(A)]

= [log_C(B)/log_C(A)][log_C(A)/log_C(B)] . . . . .Change-of-base

= 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Multiplication and Division cancel

= log_C(C). . . . . . . . . . . . . . . . . . . . . . . . . .log_a(a) = 1 for all a ≠ 0 and a ≠ 1.

= RHS

I hope this helps!

Nice name!

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