Problem 64 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \ln x=5 $$

Step-by-Step Solution

Answer

The exact solution is \( x = e^5 \); approximately, \( x \approx 148.4132 \).

Step 1: Understand the Equation

We are given the equation \( \ln x = 5 \). This is a logarithmic equation where \( \ln \) denotes the natural logarithm, which means the logarithm to the base \( e \), where \( e \approx 2.71828 \). Our task is to find the value of \( x \) that satisfies this equation.

Step 2: Use the Exponential Form

To solve \( \ln x = 5 \), we convert the equation into its exponential form. The exponential form of the equation is \( x = e^5 \). This is because the natural logarithm function, \( \ln \), is the inverse of the exponential function with base \( e \).

Step 3: Calculate the Exact Solution

The exact solution for \( x \) is \( x = e^5 \). Here, \( e^5 \) represents the exponential expression of \( e \) raised to the power of 5.

Step 4: Approximate the Solution

To find an approximate value, we calculate \( e^5 \) using a calculator. This gives us the value \( e^5 \approx 148.4132 \), rounded to four decimal places.

Key Concepts

Natural Logarithm

The natural logarithm is a crucial concept in mathematics, often denoted by \( \ln \). It is a logarithm with the base \( e \), where \( e \approx 2.71828 \), a fundamental constant in mathematics known as Euler's number. The natural logarithm is the inverse function of the exponential function. This means that if you have an equation such as \( \ln(x) = y \), it tells you the power to which \( e \) must be raised to get \( x \).
This is essential in solving problems involving logarithmic equations, as it allows transformation back and forth between logarithmic and exponential forms, making it easier to handle complex equations.
Working with the natural logarithm, you'll often see equations involving \( \ln(x) \), which means "the natural log of \( x \)." It's a powerful tool in calculus and exponential growth modeling, appearing in natural systems like population dynamics, financial calculations, and many more areas.

Exponential Form

When dealing with equations such as \( \ln(x) = 5 \), converting this logarithmic equation into its exponential form is a strategic step towards finding the solution. The exponential form of this equation would be \( x = e^5 \).
This conversion is grounded in the relationship between natural logarithms and exponents: one is the inverse of the other. By rewriting the logarithm as an exponent, we can more directly solve for \( x \).
In exponential form, \( e^y \) can be thought of as the result of multiplying the base \( e \) by itself \( y \) number of times. Thus, \( x = e^5 \) implies that \( x \) equals \( e \) raised to the 5th power, translating the logarithmic expression into something calculable and straightforward.

Approximate Solutions

Once you have expressed a logarithmic equation in exponential form, like \( x = e^5 \), finding approximate solutions becomes a matter of calculation. Calculators or computational tools can easily provide the value of \( e^5 \).
The term "approximate solution" refers to finding a numerical answer that is close to the exact value, which in some cases, like \( e^5 \), is an irrational number (it goes on indefinitely without repeating). For practical purposes, we find an approximation - here, \( e^5 \approx 148.4132 \) rounded to four decimal places.
This process is especially useful in real-world applications where exact values are difficult or impossible to measure, and approximate values provide a useful degree of precision.