Problem 247 - Calculus Volume 1

Question

Write the equation in equivalent exponential form. \(\log _{8} 2=\frac{1}{3}\)

Step-by-Step Solution

Answer

The equivalent exponential form is \( 8^{\frac{1}{3}} = 2 \).

Step 1: Understanding the Logarithmic Form

The given equation is in the logarithmic form: \( \log_{8} 2 = \frac{1}{3} \). This means we need to express the number 2 as an exponential equation with base 8, which results in \( \frac{1}{3} \).

Step 2: Converting to Exponential Form

To convert the logarithmic equation \( \log_{8} 2 = \frac{1}{3} \) to its exponential form, recall that \( \log_{b} a = c \) implies \( b^c = a \). Here, \( b = 8 \), \( a = 2 \), and \( c = \frac{1}{3} \).

Step 3: Writing the Exponential Equation

Using the formula \( b^c = a \), we substitute to get: \( 8^{\frac{1}{3}} = 2 \). This is the equivalent exponential form of the given logarithmic equation.

Key Concepts

Logarithmic Equation

A logarithmic equation is a mathematical expression that utilizes logarithms, allowing you to determine the exponent needed to raise a base to reach a certain value. In simpler terms, it helps you figure out how many times you need to multiply the base by itself to achieve a given number. For example, in the equation

• \( \log_{8} 2 = \frac{1}{3} \), the base is 8.
• The result we want to reach is 2, and the equation reveals that we need to raise 8 to the power of \( \frac{1}{3} \) in order to get 2.

Understanding logarithmic equations is crucial, as they provide a foundational tool for solving more complex mathematical problems. This process of rewriting and converting between forms, like logarithmic and exponential, offers deep insights into exponential relationships.

Base Conversion

Base conversion is vital when working with logarithms and involving switching between different bases to simplify equations. When translating a logarithmic equation such as \( \log_{8} 2 = \frac{1}{3} \), understanding base conversion is key.

• The base refers to the number you raise to some power in order to obtain a given number.
• In this case, the base is 8.
• To convert bases effectively, you often rely on properties of exponents and logarithms.

Using the change of base formula or logs, you can easily switch between bases. Base conversion is essential for accurately translating logarithmic equations into exponential form, offering a better grasp of how numbers relate through multiplication and powers.

Exponential Equation

An exponential equation is an expression involving an exponent, showcasing a relationship where a base is raised to a power. Exponential equations are the reverse of logarithmic equations: instead of solving for the exponent, you're often given the exponent.In the case of converting \( \log_{8} 2 = \frac{1}{3} \) to exponential form, you apply the knowledge that

• if \( \log_{b} a = c \), then this implies \( b^c = a \).
• This means, here
  • \(8^{\frac{1}{3}} = 2\).

Exponential equations are commonly used in fields involving exponential growth, decay, and more complex scientific phenomena. Mastery of writing exponential equations from logarithmic expressions is a powerful skill, providing clarity in deciphering growth patterns and relationships in various applications.