Problem 18 - Precalculus

Question

In Exercises \(1-33,\) solve the equation analytically. $$ 30-6 e^{-0.1 x}=20 $$

Step-by-Step Solution

Answer

\(x \approx 14.62\)

Step 1: Isolate the Exponential Term

Start with the given equation: \(30 - 6e^{-0.1x} = 20\). First, isolate the exponential term by subtracting 20 from both sides of the equation: \(10 - 6e^{-0.1x} = 0\).

Step 2: Simplify the Equation

Now, isolate \(-6e^{-0.1x}\) by subtracting 10 from both sides: \(-6e^{-0.1x} = -10\). Divide both sides by \(-6\) to further simplify: \(e^{-0.1x} = \frac{5}{3}\).

Step 3: Take the Natural Logarithm of Both Sides

To solve for \(x\), take the natural logarithm of both sides of the equation to eliminate the exponential function: \(\ln(e^{-0.1x}) = \ln\left(\frac{5}{3}\right)\).

Step 4: Use Properties of Logarithms

Apply the property \(\ln(e^y) = y\) on the left side: \(-0.1x = \ln\left(\frac{5}{3}\right)\).

Step 5: Solve for x

Finally, solve for \(x\) by dividing both sides by \(-0.1\): \[x = -\frac{\ln\left(\frac{5}{3}\right)}{0.1}\]. Evaluate this expression for the numerical value of \(x\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm is a specific type of logarithm that uses the number **e** as its base, where **e** is approximately equal to 2.718. It's commonly denoted as **ln**.
This notation is used because the natural logarithm has properties that make it especially useful in calculus and other branches of mathematics, including solving exponential equations.

• When you see **ln(e^x)**, it simplifies to **x**, because the logarithm is the inverse of the exponential function with the same base.
• Similarly, **ln(1)** is always equal to 0 because any number to the power of 0 is 1.

This natural logarithm property, where **ln(e^y) = y**, helps us remove the variable from the exponent when we're dealing with exponential equations. It makes solving these equations a more straightforward process.

Exploring Properties of Logarithms

Logarithms have unique properties that are incredibly useful for simplifying complex equations. Here are a few key properties to keep in mind:

• **Product Rule:** The logarithm of a product is the sum of the logarithms: \(\ln(ab) = \ln(a) + \ln(b)\).
• **Quotient Rule:** The logarithm of a quotient is the difference of the logarithms: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).
• **Power Rule:** The logarithm of a number raised to an exponent is the exponent times the logarithm of the base number: \(\ln(a^b) = b\ln(a)\).

These properties allow us to manipulate and simplify equations to more easily solve for unknown variables. In our solution, we used the inverse property **\(\ln(e^y) = y\)** to simplify the equation and isolate the variable **x**.

What is an Exponential Function?

An exponential function is a mathematical function of the form \(f(x) = a \cdot e^{bx}\), where **e** is Euler's number, and **a** and **b** are constants. Exponential functions are significant because they describe growth or decay processes in fields like biology, finance, and physics.
In the context of solving equations, exponential functions often lead to a scenario where the variable of interest is in the exponent. This makes them a bit tricky to handle without logarithms.

• One main characteristic is that the rate of change of an exponential function is proportional to its current value.
• This means exponential growth accelerates rapidly, which is common in compounding scenarios.

In the exercise above, we dealt with a negative exponent, indicating a form of exponential decay, along with learning how to handle such expressions by converting them with logarithms for easier manipulation.