If `"log"_(10) 2 = 0.3010, "then log"_(5) 64=`

Options

A. `(602)/(233)`
B. `(233)/(602)`
C. `(202)/(633)`
D. `(633)/(202)`

Answer

To solve the problem, we need to find the value of \( \log_5 64 \) given that \( \log_{10} 2 = 0.3010 \).

### Step-by-Step Solution:

1. **Use the Change of Base Formula:**
We can express \( \log_5 64 \) using the change of base formula:
\[ \log_5 64 = \frac{\log_{10} 64}{\log_{10} 5} \]

**Hint:** Remember that the change of base formula allows you to convert logarithms to a different base.

2. **Express 64 in Terms of Powers of 2:**
Notice that \( 64 = 2^6 \). Therefore, we can rewrite \( \log_{10} 64 \):
\[ \log_{10} 64 = \log_{10} (2^6) = 6 \cdot \log_{10} 2 \]

**Hint:** Use the property of logarithms that states \( \log_b (a^n) = n \cdot \log_b a \).

3. **Substitute the Known Value:**
Now substitute \( \log_{10} 2 = 0.3010 \) into the equation:
\[ \log_{10} 64 = 6 \cdot 0.3010 = 1.806 \]

**Hint:** Make sure to multiply correctly to find the value of \( \log_{10} 64 \).

4. **Express \( \log_{10} 5 \):**
We can express \( \log_{10} 5 \) using the relationship \( 5 = \frac{10}{2} \):
\[ \log_{10} 5 = \log_{10} 10 - \log_{10} 2 = 1 - 0.3010 = 0.6990 \]

**Hint:** Use the property \( \log_b (x/y) = \log_b x - \log_b y \) to find \( \log_{10} 5 \).

5. **Substitute Back into the Change of Base Formula:**
Now we can substitute \( \log_{10} 64 \) and \( \log_{10} 5 \) back into the formula:
\[ \log_5 64 = \frac{1.806}{0.6990} \]

**Hint:** Ensure you are dividing the two values accurately.

6. **Calculate the Final Value:**
Performing the division:
\[ \log_5 64 \approx 2.58 \]

**Hint:** Use a calculator for precise division if necessary.

### Final Answer:
Thus, \( \log_5 64 \approx 2.58 \).