Problem 30 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{8} \sqrt[3]{8}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The exponential form is \( 8^{\frac{1}{3}} = \sqrt[3]{8} \).

Step 1: Identify the Given Logarithmic Equation

We are given the logarithmic equation \( \log_{8} \sqrt[3]{8} = \frac{1}{3} \). Our task is to rewrite this equation using its equivalent exponential form.

Step 2: Understand the General Conversion Formula

Recall that a logarithmic equation of the form \( \log_{b}(x) = y \) is equivalent to the exponential equation \( b^y = x \). This is the fundamental relationship between logarithms and exponents that we'll apply.

Step 3: Apply the Conversion to the Given Equation

Using the conversion formula, the given logarithmic equation \( \log_{8} \sqrt[3]{8} = \frac{1}{3} \) can be rewritten in exponential form. Here, the base \( b \) is 8, the exponent \( y \) is \( \frac{1}{3} \), and the result \( x \) is \( \sqrt[3]{8} \). Thus, we apply this to get \( 8^{\frac{1}{3}} = \sqrt[3]{8} \).

Step 4: Verify the Exponential Form

Ensure the exponential equation makes logical sense. Given \( 8^{\frac{1}{3}} \) represents the cube root of 8, which is indeed \( \sqrt[3]{8} \), this confirms our conversion is correct.

Key Concepts

Exponential Equations: Foundations and Examples

Understanding exponential equations is fundamental to tackling logarithms and their conversions. An exponential equation is characterized by a constant base raised to a variable exponent, often written as \( b^y = x \). This can be visualized as repeated multiplication.
* If \( b = 2 \) and \( y = 3 \), then \( 2^3 = 2 \times 2 \times 2 = 8 \). * In these expressions, the base \( b \) is always a positive real number, and the exponent \( y \) can be a whole number, fraction, decimal, or even a negative number.
* Understanding each part is crucial: - **Base (b):** The number being multiplied. - **Exponent (y):** Indicates how many times the base is multiplied by itself. - **Result (x):** The outcome of the multiplication. For instance, the given exercise involves converting a logarithmic equation to its exponential form, emphasizing the relationship between these two concepts. By mastering exponential equations, you can easily switch between exponential and logarithmic forms, enhancing your overall mathematical fluency.

Logarithm to Exponent Conversion: The Bridge Between Forms

Conversion from logarithms to exponents is a fundamental skill in mathematics. It's like translating a sentence from one language to another. The standard form for this is \( \log_{b}(x) = y \), which converts to an exponential equation of the form \( b^y = x \). The steps to convert each part are straightforward:

• **Identify the base**: In the logarithm, \( b \) represents the base of the exponential equation.
• **Determine the exponent**: The result of the logarithmic expression becomes the exponent \( y \).
• **Compute the outcome**: The argument \( x \) of the logarithm is the resultant value of the exponential expression.

Considering our exercise, \( \log_{8} \sqrt[3]{8} = \frac{1}{3} \), we turn this into the exponential form \( 8^{\frac{1}{3}} = \sqrt[3]{8} \). This shows the direct relationship between the logarithmic equation and its exponential counterpart. Conversion is simply re-viewing the same operation from a different angle.

Cube Roots: Understanding Roots in Exponential Terms

Cube roots refer to a specific "opposite" operation to cubing a number, asking what number multiplied by itself three times gives a specified value. For example, the cube root of 8 is written \( \sqrt[3]{8} \), and indicates the number 2, because \( 2 \times 2 \times 2 = 8 \). This operation has a direct link with exponents, as expressing a cube root using exponents involves a fractional exponent: \( x^{\frac{1}{3}} \). To visualize:

• **Cube root equation**: \( x = 8 \) translates to \( x^{\frac{1}{3}} = \sqrt[3]{x} \).
• **For number 8**: \( 8^{\frac{1}{3}} = 2 \), confirming \( 2 \times 2 \times 2 \) results in 8.

This interplay between cube roots and exponents is crucial, allowing for simplified handling of more complex algebraic expressions. Grasping this connection makes exponential and logarithmic conversions easier, as seen in our exercise where \( 8^{\frac{1}{3}} \) rightly equals \( \sqrt[3]{8} \). Understanding cube roots deepens one’s ability to tease out useful mathematical patterns and solutions.