91Ó°ÊÓ

Exponential equations are equations where variables are found in the exponent. These types of equations often appear in situations involving growth or decay, such as in finance, biology, and physics. In our given exercise, the compound interest formula, \(A=P\left(1+\frac{r}{n}\right)^{nt}\), is an example of an exponential equation because the variable \(t\) is part of the exponent.The primary goal when dealing with exponential equations is to isolate the variable of interest, which typically involves manipulating the equation to either apply a logarithm or change it into a simpler form. This allows us to access the variable trapped in the exponent. Once the variable is isolated, solving it becomes straightforward.

Logarithms are the inverses of exponential functions and are useful in solving exponential equations. They help to "bring down" the exponent so that we can solve for the variable inside it.In the step-by-step solution provided, logarithms are applied to both sides of the equation \(\left(1+\frac{r}{n}\right)^{nt}=\frac{A}{P}\) to tackle the variable \(t\). This is done using the property of logarithms: \(\log(a^b) = b \cdot \log(a)\).

• Applying logarithms transforms the equation into \(nt \log\left(1+\frac{r}{n}\right)=\log\left(\frac{A}{P}\right)\).
• This allows us to isolate \(t\) by dividing both sides by \(n \log\left(1+\frac{r}{n}\right)\).By understanding how to use logarithms, you'll be better equipped to solve complex exponential equations.

Isolating a variable means rearranging an equation so that the variable stands alone on one side. This process is a key part of solving equations of any kind, be they linear, quadratic, or exponential.In our exercise, isolating variables was done in multiple steps:

• To solve for \(P\), the equation \(A=P\left(1+\frac{r}{n}\right)^{nt}\) was rearranged by dividing both sides by \(\left(1+\frac{r}{n}\right)^{nt}\).
• The result was \(P=\frac{A}{\left(1+\frac{r}{n}\right)^{nt}}\), neatly isolating \(P\).
• Similarly, for \(t\), the isolated term containing \(t\) was transformed using logarithms, as explained earlier, allowing for the final isolation of \(t\).

By mastering the process of isolating variables, you'll find it easier to solve a wide range of mathematical problems involving different types of functions and equations.