Question Solve for $y$ in the equation $7^{2y} = 12$ . Round the solution to the nearest hundredth.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $7^{2y} = 12$ .
2. We need to solve for $y$ .
3. We will not round any intermediate computations.
4. We will round the final answer to the nearest hundredth.

STEP 2

To solve for $y$ , we will first take the natural logarithm (ln) of both sides of the equation to utilize the property that allows us to bring the exponent down in front of the logarithm.
$\ln(7^{2y}) = \ln(12)$

STEP 3

Apply the logarithmic property $\ln(a^b) = b \cdot \ln(a)$ to the left side of the equation.
$2y \cdot \ln(7) = \ln(12)$

STEP 4

Divide both sides of the equation by $2\ln(7)$ to isolate $y$ .
$y = \frac{\ln(12)}{2\ln(7)}$

STEP 5

Now we will use a calculator to find the numerical values of $\ln(12)$ and $\ln(7)$ .
$\ln(12) \approx 2.4849$ $\ln(7) \approx 1.9459$

STEP 6

Substitute the numerical values of $\ln(12)$ and $\ln(7)$ into the equation.
$y = \frac{2.4849}{2 \cdot 1.9459}$

STEP 7

Perform the multiplication in the denominator.
$y = \frac{2.4849}{3.8918}$

STEP 8

Divide the numerator by the denominator to find the value of $y$ .
$y \approx \frac{2.4849}{3.8918} \approx 0.6385$

STEP 9

Round the answer to the nearest hundredth as instructed.
$y \approx 0.64$
The solution for $y$ in the equation $7^{2y} = 12$ is approximately $0.64$ when rounded to the nearest hundredth.

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