Solved on Jan 17, 2024

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

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

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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Solve the equation e4t+1et−2−3=10\frac{e^{4 t+1}}{e^{t-2}}-3=10et−2e4t+1​−3=10 and select the correct answer.

STEP 1

Assumptions1. The equation to solve is e4t+1et−2−3=10\frac{e^{4t+1}}{e^{t-2}} - 3 = 10et−2e4t+1​−3=10.2. We assume that ttt 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 ea/eb=ea−be^{a}/e^{b} = e^{a-b}ea/eb=ea−b.e4t+1et−2=e(4t+1)−(t−2)\frac{e^{4t+1}}{e^{t-2}} = e^{(4t+1)-(t-2)}et−2e4t+1​=e(4t+1)−(t−2)

STEP 3

Subtract the exponents.e(4t+1)−(t−2)=e4t+1−t+2e^{(4t+1)-(t-2)} = e^{4t+1-t+2}e(4t+1)−(t−2)=e4t+1−t+2

STEP 4

Combine like terms in the exponent.e4t+1−t+2=e3t+3e^{4t+1-t+2} = e^{3t+3}e4t+1−t+2=e3t+3

STEP 5

Replace the original fraction in the equation with the simplified expression.e3t+3−3=10e^{3t+3} - 3 = 10e3t+3−3=10

STEP 6

Add 3 to both sides of the equation to isolate the exponential term.e3t+3=13e^{3t+3} = 13e3t+3=13

STEP 7

Take the natural logarithm of both sides of the equation to solve for ttt. Remember that ln⁡(ex)=x\ln(e^x) = xln(ex)=x.ln⁡(e3t+3)=ln⁡(13)\ln(e^{3t+3}) = \ln(13)ln(e3t+3)=ln(13)

STEP 8

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

STEP 9

Subtract 3 from both sides of the equation to isolate the term containing ttt.3t=ln⁡(13)−33t = \ln(13) - 33t=ln(13)−3

STEP 10

Divide both sides of the equation by 3 to solve for ttt.t=ln⁡(13)−33t = \frac{\ln(13) - 3}{3}t=3ln(13)−3​

STEP 11

Calculate the value of ttt using a calculator.t≈ln⁡(13)−33t \approx \frac{\ln(13) - 3}{3}t≈3ln(13)−3​

STEP 12

Using a calculator, we find that ln⁡(13)≈2.5649\ln(13) \approx 2.5649ln(13)≈2.5649.t≈2.5649−33t \approx \frac{2.5649 - 3}{3}t≈32.5649−3​

STEP 13

Subtract 3 from 2.5649.t≈−0.43513t \approx \frac{-0.4351}{3}t≈3−0.4351​

STEP 14

Divide by 3 to get the value of ttt.t≈−0.1450t \approx -0.1450t≈−0.1450

STEP 15

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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