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Solve. Round your answer to the nearest thousandth. $\newline$ $4^x = 9$ $\newline$ $x =$ ____

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Solve exponential equations using common logarithms

Full solution

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

\newline

4^x = 9

\newline

x =

____

Identify Equation: Identify the equation to solve. $\newline$ We are given the equation $4^x = 9$ and we need to find the value of $x$ .
Apply Logarithms: Apply logarithms to both sides of the equation. $\newline$ To solve for $x$ , we can take the logarithm of both sides of the equation. We can use any logarithm, but for simplicity, we'll use the natural logarithm (ln). $\newline$ $\ln(4^x) = \ln(9)$
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. $\newline$ $x\cdot\ln(4) = \ln(9)$
Isolate x: Isolate $x$ . $\newline$ To solve for $x$ , we divide both sides of the equation by $\ln(4)$ . $\newline$ $x = \frac{\ln(9)}{\ln(4)}$
Calculate Value: Calculate the value of $x$ using a calculator. $\newline$ Using a calculator, we find: $\newline$ $x = \frac{\ln(9)}{\ln(4)}$ $\newline$ $x \approx \frac{2.197224577}{1.386294361}$ $\newline$ $x \approx 1.584962501$
Round Answer: Round the answer to the nearest thousandth. $x \approx 1.585$

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