Solved on Jan 27, 2024

Calculate $\log_6 9$ using the change of base formula, rounding the answer to the nearest thousandth.

STEP 1

Assumptions
1. We are using the change of base formula to compute $\log _{6} 9$ .
2. The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$ , where $c$ is the new base we choose to use.
3. We will use the common logarithm (base 10) for the change of base.
4. The answer should be rounded to the nearest thousandth.

STEP 2

Write down the change of base formula for logarithms.
$\log_b a = \frac{\log_c a}{\log_c b}$

STEP 3

Identify $a$ and $b$ based on the given problem.
- Here, $a = 9$ and $b = 6$ .

STEP 4

Choose a new base $c$ for the logarithm. Common choices are base 10 (common logarithm) or base $e$ (natural logarithm). We will use base 10.

STEP 5

Apply the change of base formula to $\log _{6} 9$ using base 10.
$\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6}$

STEP 6

Use a calculator to find the value of $\log_{10} 9$ .

STEP 7

Use a calculator to find the value of $\log_{10} 6$ .

STEP 8

Divide the value of $\log_{10} 9$ by the value of $\log_{10} 6$ to find the value of $\log _{6} 9$ .

STEP 9

Round the result to the nearest thousandth as per the instructions.

STEP 10

Write down the final answer.
After performing the calculations using a calculator in steps 6 to 9, you would get:
$\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6} \approx \frac{0.9542}{0.7782} \approx 1.226$
Rounded to the nearest thousandth, the answer is $1.226$ .

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Calculate $\log_6 9$ using the change of base formula, rounding the answer to the nearest thousandth.

STEP 1

STEP 2

Write down the change of base formula for logarithms.
$\log_b a = \frac{\log_c a}{\log_c b}$

STEP 3

Identify $a$ and $b$ based on the given problem.
- Here, $a = 9$ and $b = 6$ .

STEP 4

Choose a new base $c$ for the logarithm. Common choices are base 10 (common logarithm) or base $e$ (natural logarithm). We will use base 10.

STEP 5

Apply the change of base formula to $\log _{6} 9$ using base 10.
$\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6}$

STEP 6

Use a calculator to find the value of $\log_{10} 9$ .

STEP 7

Use a calculator to find the value of $\log_{10} 6$ .

STEP 8

Divide the value of $\log_{10} 9$ by the value of $\log_{10} 6$ to find the value of $\log _{6} 9$ .

STEP 9

Round the result to the nearest thousandth as per the instructions.

STEP 10

Write down the final answer.
After performing the calculations using a calculator in steps 6 to 9, you would get:
$\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6} \approx \frac{0.9542}{0.7782} \approx 1.226$
Rounded to the nearest thousandth, the answer is $1.226$ .

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Calculate log⁡69\log_6 9log6​9 using the change of base formula, rounding the answer to the nearest thousandth.

STEP 1

STEP 2

Write down the change of base formula for logarithms.log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}logb​a=logc​blogc​a​

STEP 3

Identify aaa and bbb based on the given problem.- Here, a=9a = 9a=9 and b=6b = 6b=6.

STEP 4

Choose a new base ccc for the logarithm. Common choices are base 10 (common logarithm) or base eee (natural logarithm). We will use base 10.

STEP 5

Apply the change of base formula to log⁡69\log _{6} 9log6​9 using base 10.log⁡69=log⁡109log⁡106\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6}log6​9=log10​6log10​9​

STEP 6

Use a calculator to find the value of log⁡109\log_{10} 9log10​9.

STEP 7

Use a calculator to find the value of log⁡106\log_{10} 6log10​6.

STEP 8

Divide the value of log⁡109\log_{10} 9log10​9 by the value of log⁡106\log_{10} 6log10​6 to find the value of log⁡69\log _{6} 9log6​9.

STEP 9

Round the result to the nearest thousandth as per the instructions.

STEP 10

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Calculate $\log_6 9$ using the change of base formula, rounding the answer to the nearest thousandth.

Write down the change of base formula for logarithms.
$\log_b a = \frac{\log_c a}{\log_c b}$

Identify $a$ and $b$ based on the given problem.
- Here, $a = 9$ and $b = 6$ .

Choose a new base $c$ for the logarithm. Common choices are base 10 (common logarithm) or base $e$ (natural logarithm). We will use base 10.

Apply the change of base formula to $\log _{6} 9$ using base 10.
$\log _{6} 9 = \frac{\log_{10} 9}{\log_{10} 6}$

Use a calculator to find the value of $\log_{10} 9$ .

Use a calculator to find the value of $\log_{10} 6$ .

Divide the value of $\log_{10} 9$ by the value of $\log_{10} 6$ to find the value of $\log _{6} 9$ .