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 of finding \( \log_5 64 \) given that \( \log_{10} 2 = 0.3010 \), we can use the change of base formula for logarithms. The change of base formula states that:

\[ \log_a b = \frac{\log_c b}{\log_c a} \]

where \( c \) is any positive number (commonly 10 or e). We will use base 10 for our calculations.

### Step 1: Apply the change of base formula
Using the change of base formula, we can express \( \log_5 64 \) as:

\[ \log_5 64 = \frac{\log_{10} 64}{\log_{10} 5} \]

### Step 2: Simplify \( \log_{10} 64 \)
Next, we need to simplify \( \log_{10} 64 \). We know that \( 64 = 2^6 \), so we can use the property of logarithms that states \( \log_a (b^c) = c \cdot \log_a b \):

\[ \log_{10} 64 = \log_{10} (2^6) = 6 \cdot \log_{10} 2 \]

### Step 3: Substitute the value of \( \log_{10} 2 \)
Now we can substitute the given value of \( \log_{10} 2 \):

\[ \log_{10} 64 = 6 \cdot 0.3010 = 1.806 \]

### Step 4: Find \( \log_{10} 5 \)
Next, we need to find \( \log_{10} 5 \). We can use the fact that \( 5 = \frac{10}{2} \):

\[ \log_{10} 5 = \log_{10} \left(\frac{10}{2}\right) = \log_{10} 10 - \log_{10} 2 = 1 - 0.3010 = 0.6990 \]

### Step 5: Substitute back into the formula
Now we can substitute \( \log_{10} 64 \) and \( \log_{10} 5 \) back into our formula for \( \log_5 64 \):

\[ \log_5 64 = \frac{\log_{10} 64}{\log_{10} 5} = \frac{1.806}{0.6990} \]

### Step 6: Calculate the final value
Now we perform the division:

\[ \log_5 64 \approx 2.58 \]

Thus, the final answer is approximately \( 2.58 \).

### Summary of Steps:
1. Apply the change of base formula.
2. Simplify \( \log_{10} 64 \) using the power property of logarithms.
3. Substitute the known value of \( \log_{10} 2 \).
4. Find \( \log_{10} 5 \) using the relationship with base 10.
5. Substitute back into the formula and calculate.