Question

Which expression can be used to approximate the given expression, for all positive numbers a, b, and x , where a ≠ 1 and b ≠ 1 ?

a) logₐ ⋅ log_b/log_b
b) log_b a/logₓ b
c) logₐ b/logₓ b
d) logₐ x/log_b x

Answer 1

The expression that can be used to approximate the given expression, for all positive numbers a, b, and x, is d) logₐ x/log_b x. This can be derived using the change of base formula and logarithm properties.

Step-by-step explanation:

The expression that can be used to approximate the given expression, for all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1, is d) logₐ x/log_b x. To understand why, let's break down the expression. The logarithm with base a of x is denoted as logₐ x, and the logarithm with base b of x is denoted as log_b x. The given expression logₐ x/log_b x represents the ratio of the logarithms with different bases, which is used to approximate the original expression.