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~~ ◻

Isolate exponential term: Isolate the exponential term. Start with the equation -5*e^(10t) = -30. Divide both sides by -5 to isolate the exponential term. e^(10t) = -30 / -5 e^(10t) = 6Take natural logarithm: Take the natural logarithm of both sides. To solve for t, take the natural logarithm (ln) of both sides of the equation. ln(e^(10t)) = ln(6)Apply logarithm property: Apply the property of logarithms. Use the property of logarithms that ln(e^x) = x to simplify the left side of the equation. 10t = ln(6)Solve for t: Solve for t. Divide both sides by 10 to solve for t. t = ln(6) / 10Approximate value of t: Approximate the value of t. Use a calculator to find the approximate value of ln(6) and then divide by 10. t ≈ ln(6) / 10 t ≈ 0.179/10 t ≈ 0.018 However, rounding to the nearest thousandth, we get: t ≈ 0.018

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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 $-5 \cdot e^{10 t}=-30$ . $\newline$ 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. $\newline$ $t \approx \square$ $\newline$

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Math Problems

Algebra 2

Convert between exponential and logarithmic form: all bases

Full solution

Q. Consider the equation

-5 \cdot e^{10 t}=-30

\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 \square

\newline

Isolate exponential term: Isolate the exponential term. $\newline$ Start with the equation $-5e^{10t} = -30$ . $\newline$ Divide both sides by $-5$ to isolate the exponential term. $\newline$ $e^{10t} = \frac{-30}{-5}$ $\newline$ $e^{10t} = 6$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for $t$ , take the natural logarithm ( $\ln$ ) of both sides of the equation. $\newline$ $\ln(e^{10t}) = \ln(6)$
Apply logarithm property: Apply the property of logarithms. $\newline$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the left side of the equation. $\newline$ $10t = \ln(6)$
Solve for t: Solve for t. $\newline$ Divide both sides by $10$ to solve for $t$ . $\newline$ $t = \frac{\ln(6)}{10}$
Approximate value of $t$ : Approximate the value of $t$ . Use a calculator to find the approximate value of $\ln(6)$ and then divide by $10$ . $t \approx \ln(6) / 10$ $t \approx 0.179/10$ $t \approx 0.018$ However, rounding to the nearest thousandth, we get: $t \approx 0.018$