Problem 94 - Precalculus : Building Concepts and Connections

Question

Solve using any method, and eliminate extraneous solutions. $$\ln (\log x)=1$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 10^e\)

Step 1: Remove the Natural Logarithm

This can be achieved by exponentiating both sides of the equation. Taking \(e\) to the power of both sides to cancel out the natural logarithm results in: \[e^{\ln (\log x)} = e^1\] This simplifies to \[\log x = e\]

Step 2: Solve for x

The equation \(\log x = e\) is in the form of a common logarithm, \(\log_{10}(x)\). So to isolate \(x\), you raise 10 to the power of both sides: \[10^{\log x} = 10^e\] This results in: \[x = 10^e\]

Step 3: Verify That The Solution Is Not Extraneous

Substitute \(x = 10^e\) into the original equation and verify that it holds true. First compute \(\log x\), then compute \(\ln(\log x)\) and ensure it equals 1. As long as \(x\) is greater than 0, both logarithms are defined and the equation holds. However, if \(x\) is less than or equal to 0, it will be rejected as it wouldn't make sense in the original equation (extraneous solution).

Key Concepts

Understanding Natural Logarithm

The natural logarithm is a special type of logarithm. It is denoted as \( \ln(x) \) and uses the constant \( e \) (approximately 2.718) as its base. Natural logarithms are frequently used in mathematics due to their natural association with exponential growth processes, which often appear in various scientific disciplines.

Here are some key points to remember about the natural logarithm:

• Base \( e \): The base \( e \) is an irrational and transcendental number. It’s approximately 2.718, and it emerges naturally in calculations involving continuous growth or compounding.
• Inverse Function: The natural logarithm is the inverse of the exponential function \( e^x \). This means that \( \ln(e^x) = x \) and \( e^{\ln x} = x \).
• Domain: The natural logarithm is only defined for positive numbers. That means when you see \( \ln(x) \), \( x \) must be greater than 0.

When solving equations like \( \ln(\log x)=1 \), the first step generally involves removing \( \ln \, \) which is done by exponentiating both sides of the equation with base \( e \). This step allows you to handle the common logarithm in the expression who’s inside the natural logarithm. Understanding how the natural logarithm functions and its properties can greatly assist in solving equations and understanding their solutions.

Diving into Common Logarithm

The common logarithm, denoted as \( \log(x) \) or more specifically \( \log_{10}(x) \), uses 10 as its base. This is a standard logarithm most people are familiar with. It is widely used in scientific calculations due to its relation to the decimal number system. For example, when you see \( \log x = y \), it means that \( 10^y = x \).

Important aspects of the common logarithm include:

• Base 10: Being the logarithm with base 10, it makes calculations intuitive to perform and interpret since 10 is the base of our decimal system.
• Positive Domain: Much like the natural logarithm, the common logarithm is only defined for positive values of \( x \). If \( x \leq 0 \), \( \log x \) is undefined.
• Inverse of Powers of 10: Because the common logarithm uses base 10, it reverses power operations with base 10. For instance, \( 10^{\log x} = x \).

In solving the equation \( \log x = e \), once \( \ln \) has been eliminated, it leads us to use powers of 10 to find \( x \). Notably, solving this gives \( x = 10^e \), which makes use of the property \( 10^{\log x} = x \). Gaining expertise with the common logarithm is critical for navigating and interpreting many problems in mathematics and the sciences.

Handling Extraneous Solutions

Extraneous solutions can often arise in solving logarithmic equations. An extraneous solution is a result that emerges during the process of solving an equation but does not actually satisfy the original equation.

How do extraneous solutions come about?

• They often occur when we square both sides of an equation or use logarithmic identities that assume positive numbers.
• Especially with logarithms, restrictions exist on the domains, which can cause unintended results during manipulations.

When dealing with the solution \( x = 10^e \) from the equation \( \ln(\log x) = 1 \), it is critical to check this solution fits with the original problem, ensuring both \( \ln \) and \( \log \) remain defined. This involves substituting \( x \) back into the equation and ensuring that the original statement holds true.

Here’s how checking is done:

• Evaluate \( \log x \) and substitute back into \( \ln(\log x) \).
• Verify that all terms are defined (i.e., \( x > 0 \) ensuring both logs are defined).
• Confirm that the outcome equals the stated result (in this case, \( 1 \)), affirming it is not extraneous.

By understanding and recognizing extraneous solutions, you can ensure your final solutions are valid and applicable to the original equation context.