Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. A = P (1 + r/n)^(tn), for t

Verified step by step guidance

1. Start by isolating the term that contains the variable \( t \). Divide both sides of the equation by \( P \) to get \( \frac{A}{P} = (1 + \frac{r}{n})^{tn} \).
2. To solve for \( t \), take the natural logarithm (or logarithm with any base) of both sides: \( \ln\left(\frac{A}{P}\right) = \ln\left((1 + \frac{r}{n})^{tn}\right) \).
3. Apply the power rule of logarithms, which states that \( \ln(a^b) = b \cdot \ln(a) \), to the right side: \( \ln\left(\frac{A}{P}\right) = tn \cdot \ln\left(1 + \frac{r}{n}\right) \).
4. Solve for \( t \) by dividing both sides by \( n \cdot \ln\left(1 + \frac{r}{n}\right) \): \( t = \frac{\ln\left(\frac{A}{P}\right)}{n \cdot \ln\left(1 + \frac{r}{n}\right)} \).
5. Now, you have expressed \( t \) in terms of \( A \), \( P \), \( r \), and \( n \) using logarithms.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the context of the equation A = P (1 + r/n)^(tn), the term (1 + r/n) represents the growth factor, and tn is the exponent that indicates how many times the growth factor is applied over time. Understanding how these functions behave is crucial for solving equations involving growth and decay.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an equation. When we have an equation in the form of A = P (1 + r/n)^(tn), taking the logarithm of both sides can help isolate the variable t. Logarithms can be expressed in different bases, and knowing how to convert between them is essential for solving equations effectively.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for that specific variable. In the equation A = P (1 + r/n)^(tn), we need to manipulate the equation to express t in terms of A, P, r, and n. This process often requires using algebraic techniques such as division, multiplication, and applying logarithmic properties to achieve the desired form.