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

Verified step by step guidance

1. Identify the given formula: \(A = P (1 + \frac{r}{n})^{tn}\), where \(A\) is the amount, \(P\) is the principal, \(r\) is the interest rate, \(n\) is the number of times interest is compounded per year, and \(t\) is the time in years. We need to solve for \(t\).
2. Start by isolating the exponential term. Divide both sides of the equation by \(P\) to get: \(\frac{A}{P} = (1 + \frac{r}{n})^{tn}\).
3. To solve for \(t\), take the logarithm of both sides. Since the base of the exponent is \(1 + \frac{r}{n}\), use the logarithm with this base: \(\log_{(1 + \frac{r}{n})} \left( \frac{A}{P} \right) = \log_{(1 + \frac{r}{n})} \left( (1 + \frac{r}{n})^{tn} \right)\).
4. Apply the logarithmic identity \(\log_b (b^x) = x\) to simplify the right side: \(\log_{(1 + \frac{r}{n})} \left( \frac{A}{P} \right) = tn\).
5. Finally, solve for \(t\) by dividing both sides by \(n\): \(t = \frac{1}{n} \log_{(1 + \frac{r}{n})} \left( \frac{A}{P} \right)\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as (1 + r/n)^(tn). Solving for a variable in the exponent requires techniques like logarithms to rewrite the equation in a solvable form.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and allow us to solve equations where the variable is an exponent. Understanding properties like log_b(a^c) = c log_b(a) is essential for isolating variables in exponential expressions.

Compound Interest Formula

The formula A = P(1 + r/n)^(tn) calculates compound interest, where A is the amount, P is the principal, r is the interest rate, n is the number of compounding periods per year, and t is time. Recognizing this formula helps identify which variable to solve for.