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

t~~

Consider the equation $\newline$ $-5e^{10t} = -30$ . Solve the equation for $t$ . Express the solution as a logarithm in base- $e$ . $\newline$ $t=$ $\newline$ Approximate the value of $t$ . Round your answer to the nearest thousandth. $\newline$ $t \approx$

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Properties of logarithms: mixed review

Full solution

Q. Consider the equation

\newline

-5e^{10t} = -30

. Solve the equation for

t

. Express the solution as a logarithm in base-

e

.

\newline

t=

\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 $-5$ . $\newline$ $-5\cdot e^{10t} = -30$ $\newline$ $e^{10t} = \frac{-30}{-5}$ $\newline$ $e^{10t} = 6$
Take natural logarithm: Now, we take the natural logarithm (base $e$ ) of both sides to solve for $10t$ . $\newline$ $\ln(e^{10t}) = \ln(6)$
Simplify using property of logarithms: Using the property of logarithms that $\ln(e^x) = x$ , we can simplify the left side of the equation. $10t = \ln(6)$
Solve for t: To solve for t, we divide both sides of the equation by $10$ . $\newline$ $t = \frac{\ln(6)}{10}$
Approximate the value of t: Now we approximate the value of t using a calculator. $\newline$ $t \approx \frac{\ln(6)}{10}$ $\newline$ $t \approx 0.179$

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