Question

Which expression can be used to approximate the expression below, for all positive numbers a, b, and x, where a Not-equals 1 and b Not-equals 1?

log Subscript x Baseline x
StartFraction log Subscript b Baseline x Over log Subscript b Baseline a EndFraction
StartFraction log Subscript b Baseline a Over log Subscript b Baseline x EndFraction
StartFraction log Subscript a Baseline b Over log Subscript x Baseline b EndFraction
StartFraction log Subscript a Baseline x Over log Subscript b Baseline x EndFraction

Answer 1

I’m not sure which expression you want to “approximate” — the problem text looks garbled. I’ll rewrite the four choices in standard form and show what each equals (using change of base):

Choices:
A. (log_b x)/(log_b a) = log_a x
B. (log_b a)/(log_b x) = log_x a = 1/(log_a x)
C. (log_a b)/(log_x b) = log_a x
D. (log_a x)/(log_b x) = log_a b

So:
- A and C both equal log_a x.
- B equals log_x a (the reciprocal of log_a x).
- D equals log_a b.

If you tell me the expression you meant (for example log_a x, log_x a, or log_x x), I’ll say which choice(s) match it.