Question

evaluate the logarithm

1. log 64
7
2. log 48
7
3. log 16
7

Answer 1

To evaluate the logarithms you provided, we need to approximate the values since they cannot be simplified to exact integers. You can use the change of base formula if you don't have a calculator. The change of base formula is as follows:

\[
\log_b(a) = \frac{\log_k(a)}{\log_k(b)}
\]

where \( k \) can be any positive value (often 10 or \( e \) are used).

However, here we can find the values using a calculator or approximating them mathematically. The logarithms you want to evaluate are:

1. \( \log_7(64) \)
2. \( \log_7(48) \)
3. \( \log_7(16) \)

Let's evaluate each one:

### 1. \( \log_7(64) \)

Using the change of base formula:

\[
\log_7(64) = \frac{\log_{10}(64)}{\log_{10}(7)}
\]

Using approximate values:

- \( \log_{10}(64) \approx 1.806 \)
- \( \log_{10}(7) \approx 0.845 \)

Calculating:

\[
\log_7(64) \approx \frac{1.806}{0.845} \approx 2.136
\]

### 2. \( \log_7(48) \)

Using the change of base formula:

\[
\log_7(48) = \frac{\log_{10}(48)}{\log_{10}(7)}
\]

Using approximate values:

- \( \log_{10}(48) \approx 1.681 \)

Calculating:

\[
\log_7(48) \approx \frac{1.681}{0.845} \approx 1.988
\]

### 3. \( \log_7(16) \)

Using the change of base formula:

\[
\log_7(16) = \frac{\log_{10}(16)}{\log_{10}(7)}
\]

Using approximate values:

- \( \log_{10}(16) \approx 1.204 \)

Calculating:

\[
\log_7(16) \approx \frac{1.204}{0.845} \approx 1.422
\]

### Summary of calculated values:
1. \( \log_7(64) \approx 2.136 \)
2. \( \log_7(48) \approx 1.988 \)
3. \( \log_7(16) \approx 1.422 \)

These values are approximations.