Question Solve the equation $\frac{e^{4 t+1}}{e^{t-2}}-3=10$ and select the correct answer.

Studdy Solution

STEP 1

Assumptions
1. The equation to solve is $\frac{e^{4t+1}}{e^{t-2}} - 3 = 10$ .
2. We assume that $t$ is a real number.
3. We will use properties of exponents to simplify the equation.

STEP 2

First, we simplify the expression by using the property of exponents that states $e^{a}/e^{b} = e^{a-b}$ .
$\frac{e^{4t+1}}{e^{t-2}} = e^{(4t+1)-(t-2)}$

STEP 3

Subtract the exponents.
$e^{(4t+1)-(t-2)} = e^{4t+1-t+2}$

STEP 4

Combine like terms in the exponent.
$e^{4t+1-t+2} = e^{3t+3}$

STEP 5

Replace the original fraction in the equation with the simplified expression.
$e^{3t+3} - 3 = 10$

STEP 6

Add 3 to both sides of the equation to isolate the exponential term.
$e^{3t+3} = 13$

STEP 7

Take the natural logarithm of both sides of the equation to solve for $t$ . Remember that $\ln(e^x) = x$ .
$\ln(e^{3t+3}) = \ln(13)$

STEP 8

Apply the property of logarithms to simplify the left side of the equation.
$3t+3 = \ln(13)$

STEP 9

Subtract 3 from both sides of the equation to isolate the term containing $t$ .
$3t = \ln(13) - 3$

STEP 10

Divide both sides of the equation by 3 to solve for $t$ .
$t = \frac{\ln(13) - 3}{3}$

STEP 11

Calculate the value of $t$ using a calculator.
$t \approx \frac{\ln(13) - 3}{3}$

STEP 12

Using a calculator, we find that $\ln(13) \approx 2.5649$ .
$t \approx \frac{2.5649 - 3}{3}$

STEP 13

Subtract 3 from 2.5649.
$t \approx \frac{-0.4351}{3}$

STEP 14

Divide by 3 to get the value of $t$ .
$t \approx -0.1450$

STEP 15

Comparing the calculated value of $t$ to the multiple-choice options, we find that the closest value is option C $t=-0.145$ .
The solution to the equation $\frac{e^{4t+1}}{e^{t-2}} - 3 = 10$ is $t \approx -0.145$ , which corresponds to option C.

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