Math / Algebra

Question4.5 Exponential and Logarithmic Equations and Applications Question 8 of 26 (2 points) I Question Attempt: 2 of Unilimited Antonina $\checkmark 1$ $\checkmark 2$ $\checkmark 3$ $\checkmark 4$ $\checkmark 5$ $\checkmark 6$ $\checkmark 7$ 8 $\checkmark 9$ $\checkmark 10$ 11 Español 13
Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. $2^{t}=19$ There is no solution, $\}$ . The exact solution set is $\square$ \} $t \approx$ $\square$ \begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|l|}{} \\ \hline $\square \ln \square$ & $\square \log \square$ & log ${ }^{\text {D }}$ \\ \hline ㅁ & $\sqrt[\square]{\square}$ & $\square$ \\ \hline $\times$ & & 5 \\ \hline \end{tabular} Check Save For Later Submit Assignment (C) 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center！Accessibility Dec 3 6:56

Studdy Solution

STEP 1

1. The equation $2^t = 19$ is an exponential equation.
2. We need to express the solution in terms of logarithms.
3. We will provide both the exact solution using logarithms and an approximate numerical solution to four decimal places.

STEP 2

1. Use logarithms to solve for $t$ .
2. Express the exact solution using common or natural logarithms.
3. Calculate the approximate solution to four decimal places.

STEP 3

To solve for $t$ , take the natural logarithm (ln) of both sides of the equation:
$\ln(2^t) = \ln(19)$

STEP 4

Use the logarithmic identity $\ln(a^b) = b \ln(a)$ to bring down the exponent:
$t \ln(2) = \ln(19)$

STEP 5

Solve for $t$ by dividing both sides by $\ln(2)$ :
$t = \frac{\ln(19)}{\ln(2)}$
This is the exact solution in terms of natural logarithms.

STEP 6

To find the approximate solution, calculate the value of $t$ using a calculator:
$t \approx \frac{\ln(19)}{\ln(2)} \approx 4.2479$
The exact solution set is:
$\left\{ \frac{\ln(19)}{\ln(2)} \right\}$
The approximate solution is:
$t \approx 4.2479$

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Studdy Solution

STEP 1

1. The equation $2^t = 19$ is an exponential equation.
2. We need to express the solution in terms of logarithms.
3. We will provide both the exact solution using logarithms and an approximate numerical solution to four decimal places.

STEP 2

1. Use logarithms to solve for $t$ .
2. Express the exact solution using common or natural logarithms.
3. Calculate the approximate solution to four decimal places.

STEP 3

To solve for $t$ , take the natural logarithm (ln) of both sides of the equation:
$\ln(2^t) = \ln(19)$

STEP 4

Use the logarithmic identity $\ln(a^b) = b \ln(a)$ to bring down the exponent:
$t \ln(2) = \ln(19)$

STEP 5

Solve for $t$ by dividing both sides by $\ln(2)$ :
$t = \frac{\ln(19)}{\ln(2)}$
This is the exact solution in terms of natural logarithms.

STEP 6

To find the approximate solution, calculate the value of $t$ using a calculator:
$t \approx \frac{\ln(19)}{\ln(2)} \approx 4.2479$
The exact solution set is:
$\left\{ \frac{\ln(19)}{\ln(2)} \right\}$
The approximate solution is:
$t \approx 4.2479$

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Studdy Solution

STEP 1

1. The equation 2t=19 2^t = 19 2t=19 is an exponential equation.2. We need to express the solution in terms of logarithms.3. We will provide both the exact solution using logarithms and an approximate numerical solution to four decimal places.

STEP 2

1. Use logarithms to solve for t t t.2. Express the exact solution using common or natural logarithms.3. Calculate the approximate solution to four decimal places.

STEP 3

To solve for t t t, take the natural logarithm (ln) of both sides of the equation:ln⁡(2t)=ln⁡(19) \ln(2^t) = \ln(19) ln(2t)=ln(19)

STEP 4

Use the logarithmic identity ln⁡(ab)=bln⁡(a)\ln(a^b) = b \ln(a)ln(ab)=bln(a) to bring down the exponent:tln⁡(2)=ln⁡(19) t \ln(2) = \ln(19) tln(2)=ln(19)

STEP 5

Solve for t t t by dividing both sides by ln⁡(2)\ln(2)ln(2):t=ln⁡(19)ln⁡(2) t = \frac{\ln(19)}{\ln(2)} t=ln(2)ln(19)​This is the exact solution in terms of natural logarithms.

STEP 6

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1. The equation $2^t = 19$ is an exponential equation.
2. We need to express the solution in terms of logarithms.
3. We will provide both the exact solution using logarithms and an approximate numerical solution to four decimal places.

1. Use logarithms to solve for $t$ .
2. Express the exact solution using common or natural logarithms.
3. Calculate the approximate solution to four decimal places.

To solve for $t$ , take the natural logarithm (ln) of both sides of the equation:
$\ln(2^t) = \ln(19)$

Use the logarithmic identity $\ln(a^b) = b \ln(a)$ to bring down the exponent:
$t \ln(2) = \ln(19)$

Solve for $t$ by dividing both sides by $\ln(2)$ :
$t = \frac{\ln(19)}{\ln(2)}$
This is the exact solution in terms of natural logarithms.