Use the change-of-base formula to approximate the logarithm accurate to the nearest ten-thousandth.

log√7(4.5)

To approximate \( \log_{\sqrt{7}}(4.5) \) using the change-of-base formula, we will use the natural logarithm (ln) for convenience. The change-of-base formula is given by:

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

where \( b = \sqrt{7} \) and \( a = 4.5 \), and \( c \) can be \( e \) (natural logarithm). 

Using natural logarithms:

\[
\log_{\sqrt{7}}(4.5) = \frac{\ln(4.5)}{\ln(\sqrt{7})}
\]

First, we need to find the natural logarithms of 4.5 and \(\sqrt{7}\):

\[
\ln(4.5) \approx 1.5041
\]

\[
\ln(\sqrt{7}) = \ln(7^{1/2}) = \frac{1}{2} \ln(7) \approx \frac{1}{2} \times 1.9459 \approx 0.97295
\]

Now, we can calculate the logarithm:

\[
\log_{\sqrt{7}}(4.5) = \frac{1.5041}{0.97295} \approx 1.5459
\]

Thus, the value of \( \log_{\sqrt{7}}(4.5) \) accurate to the nearest ten-thousandth is approximately \( 1.5459 \).

Do you want details or have any questions?

Here are five numbered questions for you to ask:
1. How do you calculate natural logarithms using a calculator?
2. Why is the change-of-base formula useful?
3. Can you explain the relationship between natural logarithms and common logarithms?
4. How can logarithms be used to solve exponential equations?
5. What are the applications of logarithms in real-world scenarios?

Tip: Always ensure your calculator is set to the correct mode (degree/radian) when dealing with logarithmic calculations involving trigonometric functions.

Learn how to approximate logarithms using the change-of-base formula. Step-by-step solution for \( \log_{\sqrt{7}}(4.5) \) using natural logarithms and the formula. Suitable for high school students.

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