Question

If log\(_{10}\)(5) = a and log\(_{10}\)(3) = b, express log\(_{10}\)(75) in terms of a & b.

• A. b-a
• B. 2a+b
• C. a+2b
• D. a-b

Hint:

Use the given logs to express either $\log_{10}2$ or the powers $10^a$ and $10^b$. Rewrite $75$ using the factors $3$ and $5$.

Answer 1

The Correct Option is B

Concept:

• Two logarithms are equal when their base-$10$ powers are equal.
• Build $75$ directly from the given logarithmic values.

Step 1: Consider the proposed expression.
Let $L=2a+b$.

Step 2: Raise $10$ to this power.
$10^L=10^{2a+b}=(10^a)^2(10^b)^1=5^2\times3$.

Step 3: Identify the number.
$5^2\times3=25\times3=75$, so $10^L=75$ and $L=\log_{10}75$.

Final Answer: $2a+b$