How can I find the value of x if 3^2x+1=2^x-1?

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How can I find the value of x if 3^2x+1=2^x-1?👇🏾👇 🏾

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To find the value of $𝑥$ in the equation $3^{2 𝑥 + 1} = 2^{𝑥 - 1}$ , follow these steps:

Rewrite the Equation:
$3^{2 𝑥 + 1} = 2^{𝑥 - 1}$
Take the Natural Logarithm of Both Sides:
$\ln (3^{2 𝑥 + 1}) = \ln (2^{𝑥 - 1})$
Use the Logarithm Power Rule:
$(2 𝑥 + 1) \ln (3) = (𝑥 - 1) \ln (2)$
Distribute the Logarithms:
$2 𝑥 \ln (3) + \ln (3) = 𝑥 \ln (2) - \ln (2)$
Isolate Terms Involving $𝑥$ :
$2 𝑥 \ln (3) - 𝑥 \ln (2) = - \ln (2) - \ln (3)$
Factor Out $𝑥$ :
$𝑥 (2 \ln (3) - \ln (2)) = - \ln (2) - \ln (3)$
Solve for $𝑥$ :
$𝑥 = \frac{- \ln (2) - \ln (3)}{2 \ln (3) - \ln (2)}$

This expression gives the exact solution for $𝑥$ .

To simplify and verify, we can calculate the numerical value:

Calculate the Values of the Logarithms:
- $\ln (2) \approx 0.693$
- $\ln (3) \approx 1.099$
Substitute and Simplify:
$𝑥 = \frac{- (0.693 + 1.099)}{2 \times 1.099 - 0.693} = \frac{- 1.792}{2.198 - 0.693} = \frac{- 1.792}{1.505} \approx - 1.19$

Thus, the approximate value of $𝑥$ is $- 1.19$ .

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