Questions: Use a change of base formula to evaluate the logarithm. Give your answer rounded to four decimal places. log6 sqrt(2) log6 sqrt(2)= (Simplify your answer. Do not round until the final answer. Then round to four decimal places as needed.)

Use a change of base formula to evaluate the logarithm. Give your answer rounded to four decimal places.

log6 sqrt(2)

log6 sqrt(2)=

(Simplify your answer. Do not round until the final answer. Then round to four decimal places as needed.)

Transcript text: Use a change of base formula to evaluate the logarithm. Give your answer rounded to four decimal places. \[ \begin{array}{c} \log _{6} \sqrt{2} \\ \log _{6} \sqrt{2}= \end{array} \] $\square$ (Simplify your answer. Do not round until the final answer. Then round to four decimal places as needed.)

Solution

Solution Steps

To evaluate the logarithm using the change of base formula, we can convert the logarithm to a base that is more convenient, such as base 10 or base $ e $ (natural logarithm). The change of base formula is given by: \[ \log_b a = \frac{\log_k a}{\log_k b} \] where $ k $ is the new base (commonly 10 or $ e $).

For the given problem, we will use the natural logarithm (base $ e $) to evaluate $\log_6 \sqrt{2}$.

Solution Approach

Use the change of base formula to convert $\log_6 \sqrt{2}$ to natural logarithms.
Compute the natural logarithms using Python.
Divide the results and round the final answer to four decimal places.

Step 1: Change of Base Formula

To evaluate $ \log_6 \sqrt{2} $, we apply the change of base formula: \[ \log_6 \sqrt{2} = \frac{\log_e \sqrt{2}}{\log_e 6} \]

Step 2: Calculate $ \log_e \sqrt{2} $ and $ \log_e 6 $

We compute the natural logarithms:

$ \log_e \sqrt{2} = 0.3465735902799727 $
$ \log_e 6 = 1.791759469228055 $

Step 3: Compute the Logarithm

Now, we substitute the values into the change of base formula: \[ \log_6 \sqrt{2} = \frac{0.3465735902799727}{1.791759469228055} \approx 0.19342640361727081 \]

Step 4: Round the Result

Finally, we round the result to four decimal places: \[ \log_6 \sqrt{2} \approx 0.1934 \]