Math Snap

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.

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:
$$t \approx 4.2479$$