Consider the equation 0.5*e^(4z)=13". " 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~~

Isolate exponential term: Isolate the exponential term e^(4z). To isolate e^(4z), we need to divide both sides of the equation by 0.5. 0.5*e^(4z) = 13 e^(4z) = 13 / 0.5 e^(4z) = 26Take natural logarithm: Take the natural logarithm of both sides. To solve for z, we take the natural logarithm (ln) of both sides of the equation because the natural logarithm is the inverse function of the exponential function with base e. ln(e^(4z)) = ln(26)Apply logarithm property: Apply the property of logarithms that ln(e^x) = x. Using this property, we can simplify the left side of the equation. 4z = ln(26)Solve for z: Solve for z. To solve for z, we divide both sides of the equation by 4. z = ln(26) / 4Approximate z: Approximate the value of z. Using a calculator, we can find the approximate value of ln(26) and then divide by 4. z ≈ ln(26) / 4 z ≈ 3.2581 / 4 z ≈ 0.8145

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Consider the equation

0.5*e^(4z)=13". "
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 $\newline$ $0.5 \cdot e^{4 z}=13 \text {. }$ $\newline$ 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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Math Problems

Algebra 2

Change of base formula

Full solution

Q. Consider the equation

\newline

0.5 \cdot e^{4 z}=13 \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

Isolate exponential term: Isolate the exponential term $e^{4z}$ . $\newline$ To isolate $e^{4z}$ , we need to divide both sides of the equation by $0.5$ . $\newline$ $0.5 \cdot e^{4z} = 13$ $\newline$ $e^{4z} = \frac{13}{0.5}$ $\newline$ $e^{4z} = 26$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for $z$ , we take the natural logarithm ( $\ln$ ) of both sides of the equation because the natural logarithm is the inverse function of the exponential function with base $e$ . $\newline$ $\ln(e^{4z}) = \ln(26)$
Apply logarithm property: Apply the property of logarithms that $\ln(e^x) = x$ . $\newline$ Using this property, we can simplify the left side of the equation. $\newline$ $4z = \ln(26)$
Solve for z: Solve for z. $\newline$ To solve for z, we divide both sides of the equation by $4$ . $\newline$ $z = \frac{\ln(26)}{4}$
Approximate z: Approximate the value of z. $\newline$ Using a calculator, we can find the approximate value of $\ln(26)$ and then divide by $4$ . $\newline$ $z \approx \frac{\ln(26)}{4}$ $\newline$ $z \approx \frac{3.2581}{4}$ $\newline$ $z \approx 0.8145$