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

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Convert between exponential and logarithmic form

Full solution

Q. Consider the equation

9 \cdot e^{2 z}=54

.

\newline

Solve the equation for

z

. Express the solution as a logarithm in base-

e

.

\newline

z = \square

\newline

Approximate the value of

z

. Round your answer to the nearest thousandth.

\newline

z \approx \square

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 ln of both sides: Take the natural logarithm (logarithm base $e$ , denoted as ln) of both sides to solve for $2z$ . $\newline$ $\ln(e^{2z}) = \ln(6)$
Simplify using property: Use the property of logarithms that $\ln(e^x) = x$ to simplify the left side of the equation. $2z = \ln(6)$
Divide by $2$ : Divide both sides by $2$ to solve for $z$ . $z = \frac{\ln(6)}{2}$
Approximate value: Use a calculator to approximate the value of $z$ to the nearest thousandth. $\newline$ $z \approx \frac{\ln(6)}{2}$ $\newline$ $z \approx \frac{0.895}{2}$ $\newline$ $z \approx 0.448$

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