91Ó°ÊÓ

An exponential equation is one where the variable appears in the exponent. For example, in our exercise, we have the equation \(e^{-x} = 3.2\). Here, the variable \(x\) is in the exponent of the base \(e\), which is the natural exponent (approximately 2.71828). Exponential equations can often be solved by taking the logarithm of both sides of the equation.
Remember, the goal with exponential equations is to isolate the variable. This often involves logarithms, which are the inverse functions of exponentials. By using logarithms, we can transform the equation into a more manageable form.

Logarithms have several important properties that make them useful for solving equations. One of the key properties is that the logarithm of an exponential function returns the exponent. Specifically, for the natural logarithm \(\ln\), we have \(\ln(e^{y}) = y\).
In our exercise, we started with \(e^{-x} = 3.2\) and took the natural logarithm of both sides, resulting in \(\ln(e^{-x}) = \ln(3.2)\).
Using the logarithm property \(\ln(e^y) = y\), this simplifies to \(-x = \ln(3.2)\) because \(\ln(e^{-x}) = -x\). This property is very powerful in converting exponential forms to linear forms which can be solved more easily.
Another useful property of logarithms is the product rule: \(\ln(ab) = \ln(a) + \ln(b)\). Although it wasn't needed in this problem, it's good to know for solving more complex logarithmic equations.

To solve logarithmic equations, we use properties of logarithms to isolate the variable. Let's recap the steps used in the exercise:

• First, identify the logarithmic or exponential form. Our problem was in exponential form: \(e^{-x} = 3.2\).
• Next, apply the natural logarithm to both sides. This transforms the equation using \(\ln(e^{y}) = y\).
• Simplify using the logarithm properties. We used \(\ln(e^{-x}) = -x\) to get \(-x = \ln(3.2)\).
• Finally, solve for the variable \(x\). Here, we multiplied by \(-1\) to isolate \(x\), giving \(x = -\ln(3.2)\).

By breaking down the problem into these steps, logarithmic equations become easier to handle.
Always remember to utilize logarithmic properties to simplify your equations. With practice, solving these equations will become more intuitive.