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Consider the equation

-3*e^(5w)=-88". "
Solve the equation for
w. Express the solution as a logarithm in base
e.

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

w~~

Consider the equation $\newline$ $-3 \cdot e^{5 w}=-88 \text {. }$ $\newline$ Solve the equation for $w$ . Express the solution as a logarithm in basee. $\newline$ $w=$ $\newline$ Approximate the value of $w$ . Round your answer to the nearest thousandth. $\newline$ $w \approx$

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Solve polynomial equations

Full solution

Q. Consider the equation

\newline

-3 \cdot e^{5 w}=-88 \text {. }

\newline

Solve the equation for

w

. Express the solution as a logarithm in basee.

\newline

w=

\newline

Approximate the value of

w

. Round your answer to the nearest thousandth.

\newline

w \approx

Isolate exponential term: Isolate the exponential term. $\newline$ To solve for $w$ , we first need to isolate the exponential term $e^{5w}$ . We do this by dividing both sides of the equation by $-3$ . $\newline$ Calculation: $-3\cdot e^{5w} = -88 \,|\, :(-3)$ $\newline$ $e^{5w} = \frac{88}{3}$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for the exponent, we take the natural logarithm ( $\ln$ ) of both sides of the equation, because $\ln$ and the exponential function are inverse operations. $\newline$ Calculation: $\ln(e^{5w}) = \ln(\frac{88}{3})$
Apply property of logarithms: Apply the property of logarithms. $\newline$ Using the property of logarithms that $\ln(e^x) = x$ , we can simplify the left side of the equation. $\newline$ Calculation: $5w = \ln(\frac{88}{3})$
Solve for w: Solve for w. $\newline$ Now, we divide both sides of the equation by $5$ to solve for $w$ . $\newline$ Calculation: $w = \ln(\frac{88}{3}) / 5$
Approximate value of $w$ : Approximate the value of $w$ . Using a calculator, we can find the numerical value of $w$ . Calculation: $w \approx \frac{\ln(\frac{88}{3})}{5}$ $w \approx \frac{\ln(29.3333\ldots)}{5}$ $w \approx \frac{3.3771}{5}$ $w \approx 0.67542$ Rounded to the nearest thousandth: $w \approx 0.675$

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