Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Give the exact answer. $$\log _{x} 81=4$$

Step-by-Step Solution

Answer

The value of \( x \) is 3.

Step 1: Identify the Equation

The given logarithmic equation is \( \log_{x} 81 = 4 \). This means that \( x^4 = 81 \). We want to find the value of \( x \) that satisfies this equation.

Step 2: Rewrite the Logarithmic Equation

Convert the equation from its logarithmic form to its exponential form: \( x^4 = 81 \).

Step 3: Solve for x by Taking the Fourth Root

To solve for \( x \), take the fourth root of both sides of the equation: \( x = \sqrt[4]{81} \).

Step 4: Simplify the Fourth Root

Since 81 can be written as \( 3^4 \), taking the fourth root gives \( x = 3 \). This is because \( \sqrt[4]{3^4} = 3 \).

Key Concepts

Exponential Form

Understanding exponential form is key to solving logarithmic equations. When a logarithmic equation such as \( \log_{x} 81 = 4 \) is given, it tells us there is a hidden exponential relationship between the numbers. This equation states that when \( x \) is raised to the power of 4 (exponent), the result is 81. Here's the breakdown:

• The "base" of the logarithm is \( x \).
• The "exponent" or power, in this case, is 4.
• The "result" is 81.

By rewriting the logarithmic equation in its exponential form, we read it as \( x^4 = 81 \). This is crucial because it shifts the problem from a logarithmic perspective to an algebraic one, allowing us to solve it by other means.

Fourth Root

The concept of roots in mathematics is essential when dealing with exponential forms. Specifically, a fourth root asks the question: What number, when raised to the fourth power, equals the given number?In this scenario, we have the equation \( x^4 = 81 \). To isolate \( x \), we need to find the fourth root of 81, which is indicated by the radical symbol \( \sqrt[4]{81} \). This step is similar to "undoing" the exponent, allowing us to determine what \( x \) must be.To find the fourth root:

• Recognize that 81 is a perfect power of 3, since \( 81 = 3^4 \).
• Hence, \( \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \).

By taking the fourth root of 81, you determine that \( x = 3 \). Understanding this concept helps solve problems that involve root extraction and simplifies the process of solving exponential equations.

Simplifying Expressions

Simplifying expressions is a vital step in solving math problems and plays a significant role in reaching the final solution more easily. Simplifying often involves breaking down more complex equations into simpler forms that are easier to work with.In the given problem, after rewriting in exponential form \( x^4 = 81 \), simplifying occurs when you identify 81 as \( 3^4 \). Recognizing powers like these allows us to directly apply the operation of root extraction.Steps to simplify:

• Express 81 as \( 3^4 \) to aid root manipulation.
• Calculate \( \sqrt[4]{3^4} \); since raising a number to a power and taking the root cancel each other, you find that \( x = 3 \).

Simplifying expressions not only makes calculations more straightforward but also enhances understanding of the relationships between numbers, leading to more efficient problem-solving.