Questions: Write the expression in terms of ln : log(8)(3x+2)

Transcript text: Write the expression in terms of $\mathbf{l n}$ : $\log _{8}(3 x+2)$

Solution

Solution Steps

To express the logarithm $\log_{8}(3x+2)$ in terms of natural logarithms ($\ln$), we can use the change of base formula. The change of base formula states that $\log_{b}(a) = \frac{\ln(a)}{\ln(b)}$. Here, $b = 8$ and $a = 3x + 2$.

Step 1: Understanding the Problem

We are given the expression $\log_{8}(3x + 2)$ and need to rewrite it in terms of natural logarithms, denoted as $\ln$.

Step 2: Using the Change of Base Formula

The change of base formula for logarithms allows us to convert a logarithm of any base to a logarithm of another base. The formula is:

\[ \log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)} \]

where $k$ is any positive number. In this case, we want to convert to natural logarithms, so we use $k = e$, which gives us:

\[ \log_{8}(3x + 2) = \frac{\ln(3x + 2)}{\ln(8)} \]

Step 3: Simplifying the Expression

The expression $\frac{\ln(3x + 2)}{\ln(8)}$ is already in its simplest form in terms of natural logarithms.

Final Answer

The expression $\log_{8}(3x + 2)$ in terms of natural logarithms is:

\[ \boxed{\frac{\ln(3x + 2)}{\ln(8)}} \]

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