Question

Which expression can be used to approximate the expression below, for all positive numbers aa, bb, and xx, where a≠1a=1 and b≠1b=1?

log⁡a(log⁡bx)/log⁡a(b)⋅log⁡b(x)

a) log⁡b(a)⋅log⁡a(x)/log⁡b(x)

b) log⁡b(a)/log⁡a(b)⋅log⁡b(x)

c) log⁡a(b)⋅log⁡a(x)/log⁡b(x)

d) log⁡a(x)/log⁡a(b)⋅log⁡b(x)

Answer 1

The correct expression to approximate log₊(log₋(x)) / log₊(b) is given by log₊(x) / log₊(b) · log₋(x), using the change of base formula and the logarithm power rule.

Step-by-step explanation:

The expression that can be used to approximate loga(logb(x)) / loga(b) for all positive numbers a, b, and x, given that a ≠ 1 and b ≠ 1, can be derived using the properties of logarithms. Specifically, the change of base formula for logarithms and the property that logb(xn) = n · logb(x).

Using the change of base formula, we can rewrite loga(logb(x)) as (logb(logb(x)) / logb(a)). Also, we can use the property that the logarithm of a power is the product of the exponent and the logarithm to handle the logb(x). As such, the correctly approximated expression is loga(x) / loga(b) · logb(x), which corresponds to option (d).