Questions: Use the change of base formula to compute log base 4 of 7. Round your answer to the nearest thousandth.

Transcript text: Use the change of base formula to compute $\log _{4} 7$. Round your answer to the nearest thousandth.

Solution

Solution Steps

To solve for $\log_{4} 7$ using the change of base formula, we can convert it to a fraction of logarithms with a common base, typically base 10 or base $e$. The change of base formula is given by:

\[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]

Here, we can use base 10 (common logarithm) for simplicity.

Solution Approach

Use the change of base formula to convert $\log_{4} 7$ to $\frac{\log_{10} 7}{\log_{10} 4}$.
Compute the values of $\log_{10} 7$ and $\log_{10} 4$ using Python's math library.
Divide the two results to get the final answer.
Round the result to the nearest thousandth.

Step 1: Change of Base Formula

To compute $\log_{4} 7$, we apply the change of base formula:

\[ \log_{4} 7 = \frac{\log_{10} 7}{\log_{10} 4} \]

Step 2: Calculate Logarithm Values

We find the values of $\log_{10} 7$ and $\log_{10} 4$:

\[ \log_{10} 7 \approx 0.8451 \] \[ \log_{10} 4 \approx 0.6021 \]

Step 3: Compute the Result

Now, we substitute the values into the change of base formula:

\[ \log_{4} 7 \approx \frac{0.8451}{0.6021} \approx 1.4037 \]

Step 4: Round the Result

Finally, we round the result to the nearest thousandth:

\[ \log_{4} 7 \approx 1.404 \]