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

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.

Answer 2

Concept:

• Use $\log_{10}2=1-\log_{10}5=1-a$.
• Rewrite $75$ in a form involving $3$, $10$, and $2$.

Step 1: Find $\log_{10}2$.
Since $\log_{10}2+\log_{10}5=\log_{10}10=1$, $\log_{10}2=1-a$.

Step 2: Rewrite $75$.
$75=3\times\dfrac{100}{4}$, so $\log_{10}75=b+2-2\log_{10}2$.

Step 3: Substitute and simplify.
$b+2-2(1-a)=b+2a$.

Final Answer: $2a+b$, option B