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Solve. Round your answer to the nearest thousandth. $\newline$ $7^x = 2$ $\newline$ $x =$ __

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Solve exponential equations using common logarithms

Full solution

Q. Solve. Round your answer to the nearest thousandth.

\newline

7^x = 2

\newline

x =

__

Write Equation: Write down the equation. $\newline$ We are given the equation $7^x = 2$ . We need to solve for $x$ .
Apply Logarithm: Apply the logarithm to both sides of the equation. $\newline$ To solve for $x$ , we can use logarithms. Applying the natural logarithm ( $\ln$ ) to both sides gives us $\ln(7^x) = \ln(2)$ .
Use Power Property: Use the power property of logarithms. The power property of logarithms states that $\ln(a^b) = b\cdot\ln(a)$ . We apply this property to simplify the left side of the equation: $x\cdot\ln(7) = \ln(2)$ .
Isolate x: Isolate $x$ . $\newline$ To solve for $x$ , we divide both sides of the equation by $\ln(7)$ : $x = \frac{\ln(2)}{\ln(7)}$ .
Calculate Value: Calculate the value of $x$ . Using a calculator, we find the values of $\ln(2)$ and $\ln(7)$ and then divide them to find $x$ . $x = \frac{\ln(2)}{\ln(7)} \approx \frac{0.69314718}{1.94591015} \approx 0.356207187$
Round Answer: Round the answer to the nearest thousandth. $\newline$ Rounding the value of $x$ to the nearest thousandth gives us $x \approx 0.356$ .

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