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Consider the equation

6*e^(0.25 t)=9". "
Solve the equation for
t. Express the solution as a logarithm in base
e.

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

t~~

Consider the equation $\newline$ $6 \cdot e^{0.25 t}=9 \text {. }$ $\newline$ Solve the equation for $t$ . Express the solution as a logarithm in base $e$ . $\newline$ $t=\square$ $\newline$ Approximate the value of $t$ . Round your answer to the nearest thousandth. $\newline$ $t \approx$

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Change of base formula

Full solution

Q. Consider the equation

\newline

6 \cdot e^{0.25 t}=9 \text {. }

\newline

Solve the equation for

t

. Express the solution as a logarithm in base

e

.

\newline

t=\square

\newline

Approximate the value of

t

. Round your answer to the nearest thousandth.

\newline

t \approx

Isolate exponential term: Isolate the exponential term. $\newline$ To solve for $t$ , we first need to isolate the exponential term $e^{0.25 t}$ on one side of the equation. We do this by dividing both sides of the equation by $6$ . $\newline$ $6 \cdot e^{0.25 t} = 9$ $\newline$ $e^{0.25 t} = \frac{9}{6}$ $\newline$ $e^{0.25 t} = 1.5$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for the exponent, we take the natural logarithm (logarithm base $e$ , denoted as $\ln$ ) of both sides of the equation. $\newline$ $\ln(e^{0.25 t}) = \ln(1.5)$
Apply logarithm property: 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. $\newline$ $0.25 t = \ln(1.5)$
Solve for t: Solve for t. $\newline$ To solve for t, we divide both sides of the equation by $0.25$ . $\newline$ $t = \frac{\ln(1.5)}{0.25}$
Calculate value of t: Calculate the value of t. $\newline$ Using a calculator, we can find the approximate value of $t$ . $\newline$ $t \approx \ln(1.5) / 0.25$ $\newline$ $t \approx 1.897 / 0.25$ $\newline$ $t \approx 7.588$ $\newline$ Rounded to the nearest thousandth, $t \approx 7.588$ .

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