In Exercises 1–4, solve for t.
1. c. e^((ln 0.2)t) = 0.4

Verified step by step guidance

1. Recognize that the equation is given as \(e^{(\ln 0.2) t} = 0.4\). The goal is to solve for \(t\).
2. Recall the property of exponents and logarithms: \(e^{\ln a} = a\). This means the expression \(e^{(\ln 0.2) t}\) can be rewritten as \((e^{\ln 0.2})^t = (0.2)^t\).
3. Rewrite the equation using this property: \((0.2)^t = 0.4\).
4. To solve for \(t\), take the natural logarithm of both sides: \(\ln((0.2)^t) = \ln(0.4)\).
5. Use the logarithm power rule to bring down the exponent: \(t \cdot \ln(0.2) = \ln(0.4)\). Then isolate \(t\) by dividing both sides by \(\ln(0.2)\): \(t = \frac{\ln(0.4)}{\ln(0.2)}\).

Key Concepts

Properties of Logarithms and Exponents

Understanding how logarithms and exponents interact is essential, especially that e^(ln a) = a. This allows simplification of expressions like e^((ln 0.2)t) to (0.2)^t, making the equation easier to solve.

Solving Exponential Equations

Solving equations where the variable is in the exponent involves rewriting the equation in a comparable base or applying logarithms to isolate the variable. This process transforms the problem into a linear equation in terms of the variable.

Using Natural Logarithms to Isolate Variables

Applying the natural logarithm (ln) to both sides of an equation helps isolate the variable when it appears as an exponent. Since ln and e are inverse functions, this step simplifies the equation and allows solving for the unknown.