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Use the change-of-base theorem to find the logarithm.

[tex]\[
\begin{array}{l}
\log_8 0.66 \\
\log_8 0.66 =
\end{array}
\][/tex]

[tex]\(\square\)[/tex]

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

Sagot :

GinnyAnsweridn
Sure! Let's solve the problem step by step.

The problem is to find [tex]\(\log_8(0.66)\)[/tex].

### Step 1: Use the Change-of-Base Theorem
The change-of-base theorem states:
[tex]\[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \][/tex]
We can use any base [tex]\( k \)[/tex], but for simplicity, we'll use the natural logarithm (base e). So, in our case:

[tex]\[ \log_8(0.66) = \frac{\ln(0.66)}{\ln(8)} \][/tex]

### Step 2: Calculate the Natural Logarithms
We need to find the natural logarithms of 0.66 and 8:

[tex]\(\ln(0.66) \approx -0.4155\)[/tex]

[tex]\(\ln(8) \approx 2.0794\)[/tex]

### Step 3: Apply the Change-of-Base Formula
With the natural logarithms calculated, plug these values into the change-of-base formula:

[tex]\[ \log_8(0.66) = \frac{\ln(0.66)}{\ln(8)} = \frac{-0.4155}{2.0794} \][/tex]

### Step 4: Perform the Division
Now, divide the numbers:

[tex]\[ \frac{-0.4155}{2.0794} \approx -0.1998 \][/tex]

### Step 5: Round the Final Answer
Finally, rounding to four decimal places gives us:

[tex]\[ \log_8(0.66) \approx -0.1998 \][/tex]

So, the value of [tex]\(\log_8(0.66)\)[/tex] is approximately [tex]\(-0.1998\)[/tex].
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