In Exercises 1–4, solve for t.
1. a. e^(-0.3t) = 27

Verified step by step guidance

1. Identify the equation given: \(e^{-0.3t} = 27\).
2. To solve for \(t\), take the natural logarithm (ln) of both sides to undo the exponential function: \(\ln\left(e^{-0.3t}\right) = \ln(27)\).
3. Use the logarithm property \(\ln\left(e^x\right) = x\) to simplify the left side: \(-0.3t = \ln(27)\).
4. Isolate \(t\) by dividing both sides of the equation by \(-0.3\): \(t = \frac{\ln(27)}{-0.3}\).
5. At this point, you can evaluate \(\ln(27)\) using a calculator and then perform the division to find the value of \(t\).

Key Concepts

Exponential Functions

An exponential function has the form f(t) = a^t, where the variable is in the exponent. Understanding how to manipulate and interpret these functions is essential for solving equations like e^(-0.3t) = 27, where the unknown appears as an exponent.

Natural Logarithm (ln)

The natural logarithm is the inverse of the exponential function with base e. Applying the natural logarithm to both sides of an equation like e^(-0.3t) = 27 allows us to 'bring down' the exponent and solve for t.

Solving Exponential Equations

Solving exponential equations involves isolating the exponential term and then using logarithms to solve for the variable in the exponent. This process often requires properties of logarithms and careful algebraic manipulation.