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Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth.

6^(x+2)=4^(2x-3)

Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. $\newline$ $6^{x+2}=4^{2 x-3}$

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Find roots using a calculator

Full solution

Q. Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth.

\newline

6^{x+2}=4^{2 x-3}

Rewrite Equation: Step $1$ : Start by rewriting the equation in a more manageable form. $\newline$ $6^{(x+2)} = 4^{(2x-3)}$ $\newline$ Take the logarithm of both sides to simplify the exponents. We can use any logarithm, but let's use the natural logarithm (ln) for simplicity. $\newline$ $\ln(6^{(x+2)}) = \ln(4^{(2x-3)})$
Apply Power Rule: Step $2$ : Apply the power rule of logarithms, which states $\ln(a^b) = b\cdot\ln(a)$ . $\$x+2$ \cdot\ln( $6$ ) = $2x-3$ \cdot\ln( $4$ )\)
Expand and Form Equation: Step $3$ : Expand both sides to form a linear equation. $\newline$ $x\ln(6) + 2\ln(6) = 2x\ln(4) - 3\ln(4)$
Isolate x: Step $4$ : Rearrange the terms to isolate x on one side. $\newline$ $x\ln(6) - 2x\ln(4) = -3\ln(4) - 2\ln(6)$ $\newline$ $x(\ln(6) - 2\ln(4)) = -3\ln(4) - 2\ln(6)$
Solve for x: Step $5$ : Solve for x. $\newline$ $x = \frac{-3\ln(4) - 2\ln(6)}{\ln(6) - 2\ln(4)}$ $\newline$ Using a calculator to find the values of the logarithms and perform the division, $\newline$ $x \approx \frac{-3\cdot1.386 - 2\cdot1.792}{1.792 - 2\cdot1.386}$ $\newline$ $x \approx \frac{-4.158 - 3.584}{1.792 - 2.772}$ $\newline$ $x \approx \frac{-7.742}{-0.98}$ $\newline$ $x \approx 7.900$

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