The expression

log
6
8
was evaluated using the change of base formula with base e as 1.161. What is the value of the expression
log
6
8
when evaluated using the change of base formula with base 4?(1 point)
Responses

5.161
5.161

0.290
0.290

4.644
4.644

1.161

1 answer

To find the value of \(\log_6 8\) using the change of base formula with base 4, we can use the following change of base formula:

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

For our case, we want to evaluate \(\log_6 8\) with base 4, so we can write:

\[ \log_6 8 = \frac{\log_4 8}{\log_4 6} \]

We also have previously computed \(\log_6 8\) using base \(e\), which has the value of \(1.161\). We can express this with natural logarithms as:

\[ \log_6 8 = \frac{\ln 8}{\ln 6} \]

Using the change of base formula, we can change the base from \(e\) to 4. Thus, we can relate these logarithms:

\[ \log_4 8 = \frac{\ln 8}{\ln 4} \quad \text{and} \quad \log_4 6 = \frac{\ln 6}{\ln 4} \]

Substituting into our formula, we get:

\[ \log_6 8 = \frac{\frac{\ln 8}{\ln 4}}{\frac{\ln 6}{\ln 4}} = \frac{\ln 8}{\ln 6} \]

Since we know \(\frac{\ln 8}{\ln 6}\) from above is \(1.161\):

Now we can find \(\log_4 8\) and \(\log_4 6\):

  1. \(\log_4 8 = \log_4 (2^3) = 3 \log_4 2\)
  2. To find \(\log_4 2\), we use the change of base again: \(\log_4 2 = \frac{\ln 2}{\ln 4} = \frac{1}{2}\) (because \(2^2 = 4\)).
  3. Therefore, \(\log_4 8 = 3 \cdot \frac{1}{2} = 1.5\).

Doing the same for \(\log_4 6\):

  1. \(\log_4 6 = \frac{\ln 6}{\ln 4}\).
  2. The value of \(\ln 6\) can be found using \(\ln 6 = \ln (2 \cdot 3) = \ln 2 + \ln 3\). We can approximate this as \(0.693 + 1.099 = 1.792\).
  3. Thus, \(\log_4 6 = \frac{1.792}{1.386} \approx 1.292\).

Now we can substitute these values into our ratio for \(\log_6 8\):

\[ \log_6 8 = \frac{\log_4 8}{\log_4 6} = \frac{1.5}{1.292} \approx 1.161 \]

Hence, the value of \(\log_6 8\) when evaluated using the change of base formula with base 4 aligns with the options offered; thus, the correct answer is:

1.161