Question

Solve the exponential equation 34=3−2x
(1 point)
Responses

x=2
x is equal to 2

x=−2
x is equal to negative 2

x=4
x is equal to 4

x=3

Answer 1

a,b,c or d?

Answer 2

To solve the exponential equation \( 34 = 3^{-2x} \), we can start by rewriting it in a simpler form.

We know that \( 3^{-2x} \) can be rewritten as \( \frac{1}{3^{2x}} \). Therefore, we can rewrite the equation as:

\[
34 = \frac{1}{3^{2x}}
\]

Taking the reciprocal of both sides, we get:

\[
\frac{1}{34} = 3^{2x}
\]

Now, we can take the logarithm of both sides to solve for \( x \):

\[
\log\left(\frac{1}{34}\right) = \log(3^{2x})
\]

Using the power rule of logarithms, we can simplify the right side:

\[
\log\left(\frac{1}{34}\right) = 2x \cdot \log(3)
\]

Now, we can solve for \( x \):

\[
x = \frac{\log\left(\frac{1}{34}\right)}{2 \log(3)}
\]

Now using logarithm properties:

\[
\log\left(\frac{1}{34}\right) = -\log(34)
\]

This leads us to:

\[
x = \frac{-\log(34)}{2 \log(3)}
\]

Calculating this will give us the value for \( x \).

For the purposes of your options, we do not appear to have a direct solution among \( \{ 2, -2, 4, 3 \} \). However, we can evaluate \( x \) approximately to compare:

1. \( \log(34) \) is approximately 1.531.
2. \( \log(3) \) is approximately 0.477.

Calculating \( x \):
\[
x \approx \frac{-1.531}{2 \times 0.477} \approx \frac{-1.531}{0.954} \approx -1.606
\]

Since -1.606 is not one of your multiple-choice answers, it seems there may be a mistake in the original formulation.

None of the provided options \( x = 2, x = -2, x = 4, x = 3 \) are solutions to the equation \( 34 = 3^{-2x} \). If you need the closest solution or if the question was reformulated, reviewing the equation's context may help. Please check if there was a typo or different values expected in the problem.

Answer 3

Let's solve the equation clearly to identify the correct answer among the options you provided:

The original equation is:

\[
34 = 3^{-2x}
\]

First, rewrite it in logarithmic form or exponential form. First, take the logarithm of both sides:

\[
\log(34) = -2x \log(3)
\]

Now solve for \(x\):

\[
x = -\frac{\log(34)}{2\log(3)}
\]

Calculating each logarithm:

- \(\log(34) \approx 1.531\)
- \(\log(3) \approx 0.477\)

Substituting in:

\[
x \approx -\frac{1.531}{2 \times 0.477} \approx -\frac{1.531}{0.954} \approx -1.606
\]

Since \(-1.606\) does not match any of the options directly:

- **Option A:** \(x = 2\)
- **Option B:** \(x = -2\)
- **Option C:** \(x = 4\)
- **Option D:** \(x = 3\)

Based on the numerical value of \(x\), none of the listed options (A, B, C, or D) are correct in terms of calculation.

However, since \(x\) is negative, the closest feasible answer would be **B: \(x = -2\)**, but it should be noted that the accurate calculation gives us approximately \(-1.606\).

If you have to pick based on the provided options, choose **B: \(x = -2\)** as the most reasonable choice.