Problem 146 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation in Exercises \(144-146 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Step-by-Step Solution

Answer

The solution to the equation \( \ln (\ln x) = 0 \) is \( x = e \).

Step 1: Simplify the Equation

The logarithmic equation that needs to be solved is \( \ln (\ln x)=0 \). Since \( \ln a=0 \) when \( a=1 \), it is possible to infer that \( \ln x = 1 \).

Step 2: Solve for x

The simplified equation \( \ln x = 1 \) can be rewritten as an exponential equation: \( x = e^1 \). So, \( x = e \) which is roughly equal to 2.71828.

Step 3: Check the Solution

To verify the solution, substitute \( x = e \) in the original equation \( \ln (\ln x) = 0 \). This yields \( \ln (\ln e) = 0 \). Since \( \ln e = 1 \), the equation then becomes \( \ln 1 = 0 \), which is a true statement, confirming that the solution is correct.

Key Concepts

Solving Logarithmic Equations

Logarithmic equations like \( \ln(\ln x) = 0 \) might seem tricky at first, but understanding the core concepts can make them easier to solve. Here's a simple guide to tackling logarithmic equations.

When you come across a logarithmic equation, the goal is to simplify it so you can find the value of \( x \). In our case, let's take \( \ln(\ln x) = 0 \). We know that for a logarithm \( \ln a = 0 \), the number \( a \) must be equal to 1. Thus, we can set the equation inside the first logarithm to equal 1, giving us \( \ln x = 1 \).

This transforms the problem into solving for \( x \) itself. We turn this logarithmic problem into an exponential form, which is typically easier to handle. This process helps us convert and solve these equations effectively.

Exponential Functions

An exponential function is key to solving many logarithmic equations, including our current one. These functions are in the form \( a^x \), where \( e \) is a common base. In nature and finance, the base \( e \approx 2.71828 \) often appears, known as Euler's number.

When we have \( \ln x = 1 \), to solve for \( x \), we rewrite the equation in exponential form. Remember, the natural logarithm \( \ln \) and the exponential function \( e^x \) are inverses. So, \( \ln x = 1 \) turns into \( x = e^1 \), simplifying to \( x = e \).

This relationship is crucial in switching between logarithmic and exponential forms, making it easier to solve such equations. This method is not just limited to simple cases, but a robust tool in mathematics, science, and engineering.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a special type of logarithm with the base \( e \), where \( e \approx 2.71828 \). It's widely used due to its natural occurrence in a variety of growth processes, including population growth and compound interest.

The term "natural" stems from its frequent appearance in natural sciences and its beautiful property related to growth rates and calculus. Its inverse, the exponential function \( e^x \), enables us to solve for values inside the logarithmic function.

For instance, when \( \ln x = 1 \), solving for \( x \) means recognizing that \( x \) must be such that its \( e \) exponent equals 1. Thus, converting this logarithm back to its exponential form, \( x = e^1 \), gives us the true values we need.

Understanding how \( \ln \) relates to \( e \) is not just key for solving equations but also enriches comprehension of natural phenomena and mathematical behavior.