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Consider the equation

9*e^(2z)=54". "
Solve the equation for
z. Express the solution as a logarithm in base
e.

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

z~~

Consider the equation $\newline$ $9 \cdot e^{2 z}=54 \text {. }$ $\newline$ Solve the equation for $z$ . Express the solution as a logarithm in basee. $\newline$ $z=$ $\newline$ Approximate the value of $z$ . Round your answer to the nearest thousandth. $\newline$ $z \approx$

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Change of base formula

Full solution

Q. Consider the equation

\newline

9 \cdot e^{2 z}=54 \text {. }

\newline

Solve the equation for

z

. Express the solution as a logarithm in basee.

\newline

z=

\newline

Approximate the value of

z

. Round your answer to the nearest thousandth.

\newline

z \approx

Divide by $9$ : Divide both sides of the equation by $9$ to isolate the exponential term. $\newline$ $9e^{2z} = 54$ $\newline$ $e^{2z} = \frac{54}{9}$ $\newline$ $e^{2z} = 6$
Take natural logarithm: Take the natural logarithm (base $e$ ) of both sides to solve for $2z$ .
$\ln(e^{2z}) = \ln(6)$
$2z = \ln(6)$ because $\ln(e^x) = x$
Divide by $2$ : Divide both sides by $2$ to solve for $z$ . $\frac{2z}{2} = \frac{\ln(6)}{2}$ $z = \frac{\ln(6)}{2}$
Approximate value: Approximate the value of $z$ using a calculator. $\newline$ $z \approx \frac{\ln(6)}{2}$ $\newline$ $z \approx \frac{0.895}{2}$ $\newline$ $z \approx 0.4475$

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