Noah borrows $\$ 2000$ from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money. $\newline$ The relationship between the amount of money, $A$ , in dollars that Noah owes his father (including interest), and the elapsed time, $t$ , in years, is modeled by the following equation. $\newline$ $A=2000 e^{0.1 t}$ $\newline$ How long did it take Noah to pay off his loan if the amount he paid to his father was equal to $\$ 2450$ ? Give an exact answer expressed as a natural logarithm.

View step-by-step help

Home

Math Problems

Algebra 2

Convert between exponential and logarithmic form: all bases

Full solution

Q. Noah borrows

\$ 2000

from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money.

\newline

The relationship between the amount of money,

A

, in dollars that Noah owes his father (including interest), and the elapsed time,

t

, in years, is modeled by the following equation.

\newline

A=2000 e^{0.1 t}

\newline

How long did it take Noah to pay off his loan if the amount he paid to his father was equal to

\$ 2450

? Give an exact answer expressed as a natural logarithm.

Identify Given Information: Identify the given information and the equation that models the relationship between the amount owed and time. $\newline$ We are given the equation $A = 2000e^{0.1t}$ , where $A$ is the amount owed including interest, and $t$ is the time in years. We are also given that Noah paid off $\$2450$ .
Set Up Equation: Set up the equation with the given amount that Noah paid off. $\newline$ We need to solve for $t$ when $A = \$2450$ . $\newline$ So, we set up the equation $2450 = 2000e^{0.1t}$ .
Isolate Exponential Term: Isolate the exponential term. $\newline$ To solve for $t$ , we first need to isolate the exponential term $e^{0.1t}$ . $\newline$ We do this by dividing both sides of the equation by $2000$ . $\newline$ $\frac{2450}{2000} = e^{0.1t}$
Simplify Left Side: Simplify the left side of the equation. $\newline$ Now we simplify the fraction on the left side of the equation. $\newline$ $\frac{2450}{2000} = 1.225$ $\newline$ So, $1.225 = e^{0.1t}$ .
Take Natural Logarithm: Take the natural logarithm of both sides. $\newline$ To solve for $t$ , we take the natural logarithm ( $\ln$ ) of both sides of the equation because the natural logarithm is the inverse function of the exponential function. $\newline$ $\ln(1.225) = \ln(e^{0.1t})$
Apply Logarithm Property: Apply the property of logarithms to simplify the right side. $\newline$ Using the property of logarithms that $\ln(e^x) = x$ , we can simplify the right side of the equation. $\newline$ $\ln(1.225) = 0.1t$
Solve for t: Solve for t. $\newline$ To find $t$ , we divide both sides of the equation by $0.1$ . $\newline$ $t = \frac{\ln(1.225)}{0.1}$
Calculate t Value: Calculate the value of $t$ . Using a calculator, we find the value of $\ln(1.225)$ and then divide by $0.1$ . $t = \frac{\ln(1.225)}{0.1} \approx \frac{2.079}{0.1} \approx 20.79$
Calculate t Value (Correction): Step $8$ (Correction): Calculate the value of $t$ correctly. $\newline$ Using a calculator, we find the value of $\ln(1.225)$ and then divide by $0.1$ . $\newline$ $t = \frac{\ln(1.225)}{0.1} \approx \frac{0.204}{0.1} \approx 2.04$