Question

If \(\log_{10}3=0.4771,\ \log_{10}7=0.8450\), then the value of \(\log_3 7\) is

• A. 1.771
• B. 1.5710
• C. 1.8460
• D. 1.6842

Hint:

To convert a logarithm from one base to another, divide both logarithms using any convenient common base such as 10 or \(e\).

Answer 1

The Correct Option is A

Step 1: Note what is given.
We know $\log_{10}3 = 0.4771$ and $\log_{10}7 = 0.8450$. We must find $\log_3 7$.

Step 2: The base is different.
We are asked for a log to base $3$, but our data is in base $10$. So we need a way to switch the base.

Step 3: Use the change of base rule.
\[ \log_a b = \frac{\log_{10} b}{\log_{10} a} \]

Step 4: Apply it here.
\[ \log_3 7 = \frac{\log_{10} 7}{\log_{10} 3} \]

Step 5: Put in the values.
\[ \log_3 7 = \frac{0.8450}{0.4771} \]

Step 6: Divide to get the answer.
\[ \frac{0.8450}{0.4771} \approx 1.771 \] \[ \boxed{1.771} \]