Problem 69 - Precalculus with Calculus Previews

Question

Solve the given equation. $$ \log _{x} 81=2 $$

Step-by-Step Solution

Answer

The solution to \( \log_x 81 = 2 \) is \( x = 9 \).

Step 1: Understanding the Logarithmic Equation

We are given the logarithmic equation \( \log_x 81 = 2 \). This equation expresses that \( x \) raised to the power of 2 equals 81. In other words, \( x^2 = 81 \).

Step 2: Isolate the Base

From \( \log_x 81 = 2 \), rewrite it in its exponential form: \( x^2 = 81 \). Since \( x^2 \) indicates squaring, our task is to find the value of \( x \) that satisfies this equation.

Step 3: Solving the Exponential Equation

To find \( x \), take the square root of both sides of the equation \( x^2 = 81 \). This gives us \( x = \sqrt{81} \).

Step 4: Evaluating the Square Root

Calculate \( \sqrt{81} \). Since the square root of 81 is 9 (because 9 times 9 equals 81), we find that \( x = 9 \).

Key Concepts

Understanding Exponential Form

The concept of exponential form is crucial when dealing with logarithmic equations. In a logarithmic expression like \( \log_x 81 = 2 \), we want to express the relationship in terms of exponents. When we say \( \log_x 81 = 2 \), it's the same as saying that if \( x \) is raised to the power of 2, we get 81. This can be rewritten in exponential form as \( x^2 = 81 \).
Understanding this shift is important because it simplifies the process of solving the equation. You're moving from a logarithmic format (which might seem abstract) to a more familiar algebraic form.

• Logarithmic form: \( \log_x 81 = 2 \)
• Exponential form: \( x^2 = 81 \)

By rewriting it, you transform a log equation into one involving powers, which can be more straightforward and recognizable for solving.
The key takeaway is that logarithms answer the question: "What power do I need to raise this base to in order to get this number?"

Demystifying Square Roots

Understanding square roots is essential, especially when solving equations like \( x^2 = 81 \). A square root is essentially the reverse of squaring a number. When a number is squared, it is multiplied by itself. Therefore, the square root of 81 asks "What number, when multiplied by itself, equals 81?"
To find \( x \), we take the square root of each side of \( x^2 = 81 \). This gives us \( x = \sqrt{81} \).

• Question: What square root equals 81?
• Solution: \( 9 \times 9 \) (or \( 9^2 \)) gives 81
• Thought: Remember to include both positive and negative solutions if considering all real numbers.

In mathematical terms, the principal square root of 81 is 9. Nevertheless, keep in mind that \( x \) could mathematically be either +9 or -9, as both squared will produce 81, depending on the context of the problem.

Solving Equations Efficiently

The process of solving equations involves finding the value of the variable that makes the equation true. Once you convert a logarithmic equation like \( \log_x 81 = 2 \) into its exponential counterpart \( x^2 = 81 \), you can solve for \( x \) easily.
The essential steps involved in solving such equations are:

• Transform the equation into exponential form if starting with logs.
• Isolate the variable. Here, \( x^2 \) was already isolated as it equaled 81.
• Solve for the variable by using arithmetic operations such as square roots.
• Check your solution by substituting it back into the original equation to confirm its validity.

This systematic approach makes solving equations more approachable. For this problem, solving for \( x \) gives us \( x = 9 \) because 9 squared equals 81, thus satisfying the converted exponential equation. Always take a moment to ensure that the solution makes sense in the context of the problem.