Problem 55 - Algebra A Combined Function

Question

Solve. $$ \log _{8} x=\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Understand the Logarithmic Equation

The problem is given in logarithmic form: \( \log_{8} x = \frac{1}{3} \). This means that the base \( 8 \) raised to the power of \( \frac{1}{3} \) equals \( x \).

Step 2: Convert to Exponential Form

To solve the logarithmic equation, we convert it to its equivalent exponential form. So, \( 8^{\frac{1}{3}} = x \). This means that \( x \) is the cube root of \( 8 \).

Step 3: Calculate the Cube Root

Calculate the cube root of \( 8 \). We need to find a number that, when multiplied by itself three times, equals \( 8 \). This number is \( 2 \) since \( 2 \times 2 \times 2 = 8 \).

Step 4: Verify the Result

Verify that the value of \( x \) satisfies the original equation. Substitute \( x = 2 \) into the equation: \( \log_{8}(2) \). Rewrite \( 2 \) as \( 8^{1/3} \) which confirms that \( \log_{8}(8^{1/3}) = \frac{1}{3} \), hence the solution is correct.

Key Concepts

Exponential Form

A logarithm is the opposite of an exponent. When we talk about converting a logarithmic equation to an exponential form, we’re turning a logarithmic statement into a simpler one by expressing it in terms of a base raised to an exponent.
For instance, the logarithmic equation \( \log_{8} x = \frac{1}{3} \) can be translated into exponential form: \( 8^{\frac{1}{3}} = x \). This shows that the base \( 8 \) raised to the power of \( \frac{1}{3} \) equals \( x \). Here, \( 8 \) is our base, and \( \frac{1}{3} \) is the exponent.

• "Base": The lower number in the logarithm statement (8 in this case).


• "Exponent": The power to which the base is raised (\( \frac{1}{3} \) here).

Understanding this conversion is crucial because it simplifies the calculation process, enabling us to determine the value of \( x \) straightforwardly.

Cube Root

The concept of a cube root is fundamental in understanding exponential and logarithmic relationships. A cube root of a number \( a \) is a value that, when multiplied by itself three times (cubed), gives \( a \) as the result.
For example, in the step \( 8^{\frac{1}{3}} = x \), the exponent \( \frac{1}{3} \) indicates the cube root. Therefore, \( x \) is what we call the cube root of \( 8 \). Solving this, we find \( 2 \), since multiplying \( 2 \) by itself three times results in \( 8 \) (\( 2 \times 2 \times 2 = 8 \)).

• To find the cube root, ask: "Which number, when cubed, equals \( 8 \)?"


• The answer is \( 2 \), because it matches the criteria.

Cube roots help us navigate through transformations between logarithmic and exponential forms, simplifying complex equations effectively.

Logarithmic Conversion

Logarithmic conversion is used to transform a logarithmic equation into an exponential one, making calculations easier. The process involves identifying the base and the exponent of the logarithm and rewriting the equation accordingly.
In our example, the logarithmic equation \( \log_{8} x = \frac{1}{3} \) converts to \( 8^{\frac{1}{3}} = x \). This conversion is vital as it allows us to easily compute \( x \) as the cube root of the base, \( 8 \).

• Recognize the base and exponent in the logarithmic equation.


• Convert: base raised to exponent = value (\( x \)).

This method not only simplifies solving the equation but also strengthens understanding of both logarithmic and exponential relationships, essential skills in many areas of mathematics.