Problem 75 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log x^{2}=2 $$

Step-by-Step Solution

Answer

The exact solution is \( x = 10 \).

Step 1: Use Logarithm Properties

Given the equation is \( \log x^2 = 2 \). Since \( \log x^2 = 2\) can be rewritten using the power rule for logarithms, \( \log x^n = n \log x \), we rewrite the equation as \( 2 \log x = 2 \).

Step 2: Simplify the Equation

Divide both sides of the equation \( 2 \log x = 2 \) by 2, which simplifies to \( \log x = 1 \).

Step 3: Exponentiate to Solve for x

To solve \( \log x = 1 \), rewrite the equation in exponential form. Recall that if \( \log x = 1 \), then \( x = 10^1 \). Solve for \( x \) to get \( x = 10 \).

Step 4: Verify the Solution

Substitute \( x = 10 \) back into the original equation to verify: \( \log (10^2) = \log 100 = 2 \). The left side equals the right side, confirming \( x = 10 \) is indeed the correct solution.

Key Concepts

Logarithm Properties

Logarithms have important properties that are extremely useful when solving logarithmic equations. Understanding these properties can make complex equations much simpler to handle. Let's explore the power rule, which is particularly relevant for our exercise.

• Power Rule: The power rule for logarithms states that \( \log x^n = n \log x \). This means that when taking the logarithm of a number raised to a power, you can multiply the power by the logarithm of the base number.

In our example, we start with \( \log x^2 = 2 \). By applying the power rule, it can be rewritten as \( 2 \log x = 2 \). This simplifies the equation significantly by removing the exponent from inside the log function. Recognizing and applying these rules can break down complex equations into a more manageable form.

Exponential Functions

Exponential functions are the reverse of logarithms, acting as a tool to "undo" logarithmic expressions. If you have a logarithmic equation like \( \log x = 1 \), you can convert it into an exponential equation.

• Basic Idea: If \( \log_{b} (y) = x \), then \( y = b^x \).

In solving the equation from our exercise, when we reached \( \log x = 1 \), we changed this to the exponential form: \( x = 10^1 \). This tells us that the exponential function corresponding to the logarithm of a number reveals the base raised to the power of the log's output. Here, since our base is 10, exponentiating gives us \( x = 10 \). Transforming between logarithmic and exponential forms is a critical skill in solving equations efficiently.

Exact Solutions

Finding exact solutions is essential, especially when an approximation may lead to a misunderstanding of the problem. For logarithmic equations like the one in the exercise, it's important to provide an exact answer in base 10.
Using the initial equation \( \log x^2 = 2 \), and transforming it appropriately, we solve it step-by-step:

• Apply the logarithm properties to simplify: \( 2 \log x = 2 \)
• Simplify further: \( \log x = 1 \)
• Convert to exponential form: \( x = 10^1 \)

By verifying the solution, we substitute back into the original equation: \( \log (10^2) = 2 \). This ensures that our exact solution, \( x = 10 \), satisfies the equation completely. Exact solutions avoid errors that can occur with rounding decimals and ensure clarity in solving mathematical problems.