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:

Whenever you need to express a logarithm in terms of others, the first step is always to prime factorize the number inside the log.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the values of log\(_{10}\)(5) and log\(_{10}\)(3) in terms of variables 'a' and 'b'. We need to express log\(_{10}\)(75) using these variables.
Step 2: Key Formula or Approach:
We will use the properties of logarithms:
1. Product Rule: log(xy) = log(x) + log(y)
2. Power Rule: log(x\(^n\)) = n \(\cdot\) log(x)
Step 3: Detailed Explanation:
First, we break down the number 75 into its prime factors, specifically using the numbers we have logarithms for (3 and 5).
\[ 75 = 25 \times 3 = 5^2 \times 3 \] Now, we take the log base 10 of 75:
\[ \log_{10}(75) = \log_{10}(5^2 \times 3) \] Using the Product Rule, we can split the logarithm:
\[ \log_{10}(5^2 \times 3) = \log_{10}(5^2) + \log_{10}(3) \] Using the Power Rule on the first term:
\[ \log_{10}(5^2) + \log_{10}(3) = 2 \cdot \log_{10}(5) + \log_{10}(3) \] Finally, we substitute the given values log\(_{10}\)(5) = a and log\(_{10}\)(3) = b:
\[ 2 \cdot (a) + (b) = 2a + b \] Step 4: Final Answer:
Therefore, log\(_{10}\)(75) can be expressed as 2a + b.