In Exercises 1–4, solve for t.
2. b. e^(kt) = 10

Verified step by step guidance

1. Identify the equation given: \(e^{kt} = 10\).
2. To solve for \(t\), take the natural logarithm (ln) of both sides to utilize the property that \(\ln(e^x) = x\).
3. Apply the natural logarithm: \(\ln(e^{kt}) = \ln(10)\).
4. Simplify the left side using the logarithm property: \(kt = \ln(10)\).
5. Isolate \(t\) by dividing both sides by \(k\): \(t = \frac{\ln(10)}{k}\).

Key Concepts

Exponential Functions

An exponential function has the form f(t) = a^t, where the variable is in the exponent. In this problem, e^(kt) represents an exponential function with base e, the natural exponential constant approximately equal to 2.718. Understanding how exponential functions behave is essential for solving equations involving them.

Natural Logarithm (ln)

The natural logarithm is the inverse function of the exponential function with base e. It allows us to solve equations where the variable is in the exponent by 'undoing' the exponential. Applying ln to both sides of e^(kt) = 10 helps isolate the variable t.

Solving Exponential Equations

To solve equations like e^(kt) = 10, we use logarithms to isolate the exponent. After taking the natural logarithm of both sides, we apply algebraic manipulation to solve for t, including dividing by the constant k. This process is fundamental in calculus and related fields.