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Consider the equation

-5*e^(10 t)=-30". "
Solve the equation for
t. Express the solution as a logarithm in base
e.

t=◻++^(+x)
Approximate the value of
t. Round your answer to the nearest thousandth.

t~~

Consider the equation $\newline$ $-5 \cdot e^{10 t}=-30 \text {. }$ $\newline$ 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

-5 \cdot e^{10 t}=-30 \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: First, we need to isolate the exponential term $e^{10t}$ by dividing both sides of the equation by $-5$ . $\newline$ $-5e^{10t} = -30$ $\newline$ $e^{10t} = \frac{-30}{-5}$ $\newline$ $e^{10t} = 6$
Take natural logarithm: Next, we take the natural logarithm (base $e$ ) of both sides to solve for $t$ . $\newline$ $\ln(e^{10t}) = \ln(6)$
Simplify left side of equation: Using the property of logarithms that $\ln(e^x) = x$ , we can simplify the left side of the equation. $10t = \ln(6)$
Divide both sides: Now, we divide both sides by $10$ to solve for $t$ . $\newline$ $t = \frac{\ln(6)}{10}$
Approximate value of t: Finally, we approximate the value of t using a calculator. $\newline$ t \approx \frac{\ln( $6$ )}{ $10$ } $\newline$ t \approx \frac{ $0$ . $179$ }{ $10$ } $\newline$ t \approx $0$ . $018$
Calculate value of t: Using a calculator to find the value of $\ln(6)$ and then dividing by $10$ to get the value of $t$ .
$t \approx \frac{\ln(6)}{10}$
$t \approx \frac{1.79176}{10}$
$t \approx 0.179$

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