Problem 12 - Algebra for College Students

Question

Solve each exponential equation and express approximate solutions to the nearest hundredth. $$ e^{x}=86 $$

Step-by-Step Solution

Answer

\(x \approx 4.45\)

Step 1: Understand the Problem

The problem requires us to solve the exponential equation \(e^x = 86\) for \(x\) and express the solution rounded to the nearest hundredth. \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Apply the Natural Logarithm

To solve for \(x\), apply the natural logarithm (\(\ln\)) to both sides of the equation. This step is used because the natural logarithm is the inverse of the exponential function, making it easier to solve for the exponent.

Step 3: Simplify the Equation

By applying the natural logarithm, the equation \(e^x = 86\) becomes \(\ln(e^x) = \ln(86)\). The left side simplifies to \(x\) since \(\ln(e^x) = x\). Thus, \(x = \ln(86)\).

Step 4: Calculate and Approximate

Use a calculator to find \(\ln(86)\). The calculated value is approximately 4.4543. Round this number to the nearest hundredth, resulting in 4.45.

Key Concepts

Natural Logarithm

When dealing with exponential equations, the natural logarithm ( ln ) plays a crucial role in simplifying calculations. The natural logarithm uses the base "e", which is an irrational number approximately equal to 2.71828.
It is the inverse function of the exponential function, and this relationship allows us to solve equations involving exponentials by transforming them into linear equations. Why is ln useful? When you have an equation like e^{x} = 86 , applying ln helps by essentially cancelling out the exponential. This is because of the property where ln(e^x) = x .

• Use ln whenever you need to "bring down the exponent" in an equation.
• Provides a straightforward path to solving equations involving the exponential constant e .

Understanding this property is essential, as it converts the problem into something that can be easily solved using basic math operations. In our case, ln(86) gives us the value x we were solving for.

Inverse Functions

Inverse functions are pairs of functions that "undo" each other. In simpler terms, if you have a function that takes an input and produces an output, its inverse will take that output back to the original input.
This concept is particularly important in solving exponential equations. When you're dealing with an equation like e^{x} = 86 , the natural logarithm serves as the inverse function. Here's how it works:

• The exponential function with base e , f(x) = e^x , maps a number x to an exponential output.
• Its inverse is ln(y) = x , which will map the exponential output back to the original value x .

The understanding of inverse functions simplifies solving equations because it provides a path to "reverse" any function, using less complex calculations. Just like we solve e^{x} = 86 by taking ln on both sides, we revert to a simple linear expression. Whenever you find a function you cannot solve easily, look for its inverse to make it work in reverse!

Rounding Numbers

Rounding numbers is an essential skill, especially when handling results from calculations like the one with ln(86) , where we obtained an approximate value for x .
To round to the nearest hundredth, you look at the third digit after the decimal point. If it is 5 or greater, you round up the second digit, and if it is less than 5, you leave the second digit as it is. Let's see this with an example:

• The number 4.4543 requires rounding.
• The third digit after the decimal is 4.
• Since 4 is less than 5, the second digit after the decimal remains 5.

Thus, 4.4543 rounded to the nearest hundredth is 4.45. This technique helps provide easier numbers to work with, making them convenient for interpretation and further calculations. Always ensure to apply the correct rounding method for the specific precision needed in your solutions.