Problem 50 - Algebra A Combined Function

Question

Solve. $$ \log _{x} 8=3 $$

Step-by-Step Solution

Answer

The value of \( x \) is 2.

Step 1: Understand the Logarithm Equation

The given equation is \( \log_{x} 8 = 3 \). This means that \( x \), raised to the power of 3, equals 8. The base \( x \) is what we need to solve for in this equation.

Step 2: Convert Logarithm to Exponential Form

To eliminate the logarithm, convert the log equation into its exponential form. This results in the equation \( x^3 = 8 \).

Step 3: Solve for x

To find \( x \), we need to solve the equation \( x^3 = 8 \). Take the cube root of both sides to get \( x = \sqrt[3]{8} \).

Step 4: Calculate the Cube Root

Calculate the cube root of 8 to find that \( \sqrt[3]{8} = 2 \). This implies that \( x = 2 \).

Key Concepts

Exponential Form

Understanding logarithms often involves converting them into exponential form. This transformation helps simplify calculations and find results more easily. For the equation \( \log_{x} 8 = 3 \), we know that the logarithmic statement represents an expression where the base \( x \), raised to a certain power, equals a given number—in this case, 8. By definition of a logarithm, this statement can be rewritten as an exponential equation, \( x^3 = 8 \). Here:

• \( x \) is the base of the logarithm and the base of the exponent in the exponential form.
• 3 is the power to which we raise \( x \).
• 8 is the result of the equation, representing the power 3 of the base \( x \).

Converting between logarithmic and exponential forms is a fundamental skill in mathematics, useful for solving equations and understanding growth patterns. Remember, \( \log_{b}(a) = c \) translates to \( b^c = a \), allowing you to switch perspectives whenever needed.

Cube Root

Once you have your equation in exponential form, the next step is often to simplify it by resolving powers. For \( x^3 = 8 \), we need to find which number, when cubed (multiplied by itself twice), results in 8. This requires finding the cube root.The cube root is the inverse operation of raising a number to the power of three. Taking the cube root of both sides of the equation \( x^3 = 8 \), we find:

• \( x = \sqrt[3]{8} \)
• \( \sqrt[3]{8} \) is asking what number, when multiplied by itself twice, results in 8.

Calculating, we find \( \sqrt[3]{8} = 2 \). This is because \( 2 \times 2 \times 2 = 8 \). Cube roots can be crucial in various mathematical scenarios, such as solving cubic equations or understanding geometric scaling, and are a basic but important tool in a mathematician's kit.

Solving Equations

Solving equations is about finding unknown values that satisfy the given mathematical statement. In our problem, after converting \( \log_{x} 8 = 3 \) into the exponential form \( x^3 = 8 \), solving means finding \( x \).The process includes:

• Understanding the relationship described by the equation—in this case, exponential growth.
• Converting forms, such as changing from logarithmic to exponential.
• Simplifying through arithmetic operations like finding roots or powers.

For \( x^3 = 8 \), when we solved it by taking the cube root of both sides, we found \( x = 2 \). This solution checks out because plugging 2 back into the original exponential form results in the correct powers. Skillfully manipulating algebraic expressions and understanding their underlying principles make solving such equations both intuitive and logical.