Problem 61 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$\ln x=1.4023$$

Step-by-Step Solution

Answer

\( x \approx 4.0649 \)

Step 1: Understand the Natural Logarithm

The problem involves the natural logarithm function, denoted as \( \ln x \). The natural logarithm is the inverse operation of taking the exponential of \( e \), where \( e \approx 2.71828 \). If \( \ln x = a \), then \( x = e^a \).

Step 2: Rewrite the Equation

We are given that \( \ln x = 1.4023 \). To solve for \( x \), we need to rewrite the equation in exponential form. This means expressing \( x \) as a power of \( e \). Thus, the rewritten form is \( x = e^{1.4023} \).

Step 3: Calculate the Power of \( e \)

Use a scientific calculator to compute \( e^{1.4023} \). Enter the value 1.4023, and then use the exponential function key to find \( e^{1.4023} \).

Step 4: Express Answer to Four Decimal Places

Once you have calculated \( e^{1.4023} \) using a calculator, you get a precise value. Round this value to four decimal places to provide the final answer.

Key Concepts

Exponential Function

The exponential function is a mathematical function typically written as \( e^x \), where \( e \) is a special constant approximately equal to 2.71828. This constant \( e \) is known as Euler's number and is the base of the natural logarithm. When we see an exponential function in this context, it means that the variable \( x \) is an exponent to which this base \( e \) is raised.
The exponential function is powerful in many real-world applications, such as modeling population growth, continuously compounded interest, and radioactive decay. It's also crucial in solving certain types of equations, especially those involving natural logarithms.
The exponential and logarithmic functions are inverse operations. This means if \( b^x = y \), then \( \log_b(y) = x \). Taking the exponential and the corresponding logarithm will cancel each other out. This relationship helps in solving equations where the variable is within a logarithm, like in the equation \( \ln x = a \).

Inverse Operations

Inverse operations are operations that reverse the effect of each other. When we talk about the natural logarithm \( \ln \) and the exponential function \( e^x \), they are inverses. This means if \( \ln(x) = y \), then \( e^y = x \) and vice versa.
Understanding inverse operations in mathematics is crucial when dealing with solving equations. For instance, in our exercise, we started with \( \ln x = 1.4023 \). By recognizing that the natural logarithm and the exponential function take each other out, we rewrote the equation to \( x = e^{1.4023} \).
Inverse operations are not just limited to exponentials and logarithms. In arithmetic, addition and subtraction are inverse operations, as are multiplication and division. Recognizing these pairs helps solve equations and simplify expressions.

Scientific Calculator

A scientific calculator is a handy tool designed to perform complex mathematical calculations. Unlike basic calculators, scientific ones include functions such as trigonometric, logarithmic, and exponential calculations.
To solve the expression \( e^{1.4023} \) using a scientific calculator, follow these steps:

• Turn on the calculator and select the function mode, typically found near other mathematical functions.
• Enter the base of the exponent, which is Euler's number \( e \). Many calculators have a dedicated \( e^x \) button.
• Input the exponent value, in this case, 1.4023.
• Press the execute or calculate button, sometimes labeled as "enter" or "equals", to find the result.
• The displayed result will be the value of \( e^{1.4023} \).

Using the correct functions on the scientific calculator ensures you get the accurate result needed for further computations.

Rounding Decimals

Rounding decimals is a method used to reduce the number of digits in a number while keeping its value close to the original. In mathematics, rounding is especially useful when dealing with long decimal numbers, making them easier to work with or report.

• To round a number to a specific decimal place, look at the digit at the place right after where you want to round to.
• If this digit is 5 or greater, round the last significant digit up. If it is less than 5, keep the last significant digit as it is.

For example, if a calculation gives you a result of 4.058932, and you need to round to four decimal places, you focus on the fifth decimal spot (in this case, 9). Since 9 is greater than 5, you would round the fourth decimal (8) up, resulting in an approximate value of 4.0589.
Rounding helps in making numbers more manageable, especially when precision beyond a certain point doesn't significantly affect the outcome of a calculation.