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Consider the equation

7*e^(0.1 t)=47". "
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 $\newline$ $7 \cdot e^{0.1 t}=47 \text {. }$ $\newline$ 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

7 \cdot e^{0.1 t}=47 \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

Identify and isolate exponential term: Identify the equation and isolate the exponential term. $\newline$ The equation is $7e^{0.1t} = 47$ . To isolate the exponential term, divide both sides of the equation by $7$ . $\newline$ $e^{0.1t} = \frac{47}{7}$ $\newline$ $e^{0.1t} = 6.71428571429$
Take natural logarithm of both sides: Take the natural logarithm of both sides to solve for $t$ . $\newline$ $\ln(e^{0.1 t}) = \ln(6.71428571429)$ $\newline$ Since $\ln(e^x) = x$ , we can simplify the left side to $0.1 t$ . $\newline$ $0.1 t = \ln(6.71428571429)$
Divide both sides by $0$ . $1$ : Divide both sides by $0.1$ to solve for $t$ . $\newline$ $t = \frac{\ln(6.71428571429)}{0.1}$
Calculate value of t: Calculate the value of t using a calculator. $\newline$ $t \approx \ln(6.71428571429) / 0.1$ $\newline$ $t \approx 19.803902575$ $\newline$ Round the answer to the nearest thousandth. $\newline$ $t \approx 19.804$

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