Question

Solve the logarithmic equation.
Logv4 (3x + 1) = 2

Answers

GPT-4o mini
To solve the logarithmic equation \( \log_{4}(3x + 1) = 2 \), we can rewrite it in its exponential form. The general form of a logarithm states that if \( \log_{b}(a) = c \), then \( a = b^c \).

Applying that to our equation:

\[
3x + 1 = 4^2
\]

Calculating \( 4^2 \):

\[
4^2 = 16
\]

Now, we have:

\[
3x + 1 = 16
\]

Next, we can solve for \( x \) by isolating it:

1. Subtract 1 from both sides:

\[
3x = 16 - 1
\]
\[
3x = 15
\]

2. Divide both sides by 3:

\[
x = \frac{15}{3}
\]
\[
x = 5
\]

So, the solution to the equation is:

\[
\boxed{5}
\]

To verify, substitute \( x = 5 \) back into the original logarithmic equation:

\[
3(5) + 1 = 15 + 1 = 16
\]
\[
\log_{4}(16) = 2
\]

Since \( 4^2 = 16 \) holds true, our solution \( x = 5 \) is correct.

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