Express log base 12 of 72 in terms of common logarithms.

• A. log(12)/log(72)
• B. log(72) − log(12)
• C. log(72)/log(144)
• D. log(72)/log(12)

Answer: (D)

This question uses the idea of changing the base of a logarithm. When you want log base b of a, you can rewrite it using common logs (base 10) as log base b of a = log a / log b, where the log on the right is any common logarithm.

So log base 12 of 72 equals log(72) divided by log(12). This is the expression you get by applying the change-of-base rule with the common logarithm.

Why this works: the ratio log a / log b measures how many times you need to multiply b to reach a, but expressed in terms of a base you already know. Using the opposite arrangement or a different denominator would correspond to a different base (like base 72 or base 144) and won’t give log base 12 of 72. Also, subtracting logs would give log(72/12), not a base change.