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Consider the equation

5*e^(-7x)=12". "
Solve the equation for
x. Express the solution as a logarithm in base
e.

x=
Approximate the value of
x. Round your answer to the nearest thousandth.

x~~

Consider the equation $\newline$ $5 \cdot e^{-7 x}=12 \text {. }$ $\newline$ Solve the equation for $x$ . Express the solution as a logarithm in basee. $\newline$ $x=$ $\newline$ Approximate the value of $x$ . Round your answer to the nearest thousandth. $\newline$ $x \approx$

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Change of base formula

Full solution

Q. Consider the equation

\newline

5 \cdot e^{-7 x}=12 \text {. }

\newline

Solve the equation for

x

. Express the solution as a logarithm in basee.

\newline

x=

\newline

Approximate the value of

x

. Round your answer to the nearest thousandth.

\newline

x \approx

Isolate exponential term: Isolate the exponential term. $\newline$ To solve for $x$ , we first need to isolate the exponential term $e^{(-7x)}$ by dividing both sides of the equation by $5$ . $\newline$ $5 \cdot e^{(-7x)} = 12$ $\newline$ $e^{(-7x)} = \frac{12}{5}$ $\newline$ $e^{(-7x)} = 2.4$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for the exponent, we take the natural logarithm (log base $e$ , also known as $\ln$ ) of both sides of the equation. $\newline$ $\ln(e^{-7x}) = \ln(2.4)$
Apply logarithmic property: Apply the logarithmic property. $\newline$ Using the property that $\ln(e^y) = y$ , we can simplify the left side of the equation. $\newline$ $-7x = \ln(2.4)$
Solve for x: Solve for x. $\newline$ Now, we divide both sides by $-7$ to solve for x. $\newline$ $x = \frac{\ln(2.4)}{-7}$
Approximate x value: Approximate the value of $x$ . Using a calculator, we can find the approximate value of $x$ . $x \approx \frac{\ln(2.4)}{-7}$ $x \approx \frac{-0.087}{-7}$ $x \approx 0.0124$ Rounded to the nearest thousandth, $x \approx 0.012$ .

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