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Solve. Round your answer to the nearest thousandth. $\newline$ $3^x = 5$ $\newline$ $x =$ ____

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Solve exponential equations using common logarithms

Full solution

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

\newline

3^x = 5

\newline

x =

____

Write Equation: Write down the equation. $\newline$ We are given the equation $3^x = 5$ . $\newline$ We need to solve for $x$ .
Apply Logarithm: Apply the logarithm to both sides of the equation. $\newline$ To solve for $x$ , we can take the logarithm of both sides. We can use any logarithm, but for simplicity, we'll use the natural logarithm (ln). $\newline$ $\ln(3^x) = \ln(5)$
Use Power Property: Use the power property of logarithms. $\newline$ 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. $\newline$ $x\cdot\ln(3) = \ln(5)$
Isolate $x$ : Isolate $x$ . $\newline$ To solve for $x$ , we divide both sides of the equation by $\ln(3)$ . $\newline$ $x = \frac{\ln(5)}{\ln(3)}$
Calculate x: Calculate the value of x using a calculator. $\newline$ Using a calculator, we find: $\newline$ $x = \frac{\ln(5)}{\ln(3)}$ $\newline$ $x \approx 1.46497352072\ldots$
Round Answer: Round the answer to the nearest thousandth. $x \approx 1.465$

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