Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 64^{\frac{1}{3}}=4 $$

Step-by-Step Solution

Answer

The logarithmic form is \( \log_{64}(4) = \frac{1}{3} \).

Step 1: Understand the Exponential Form

Given the exponential equation \(64^{\frac{1}{3}} = 4\), we identify the base as 64, the exponent as \(\frac{1}{3}\), and the result as 4. This forms the structure \( b^e = a \) where \( b = 64 \), \( e = \frac{1}{3} \), and \( a = 4 \).

Step 2: Convert to Logarithmic Form

Using the conversion from exponential to logarithmic form, we have \( b^e = a \) translates to \( \log_b(a) = e \). Here, substitute \( b = 64 \), \( a = 4 \), and \( e = \frac{1}{3} \) to get \( \log_{64}(4) = \frac{1}{3} \).

Key Concepts

Exponential Equations

Exponential equations are a type of equation where a constant base is raised to a variable exponent, resulting in a specific number. These equations appear in various mathematical and real-world scenarios, such as compound interest, population growth, or radioactive decay. Being familiar with exponential equations can help solve problems in various scientific fields.Key features of exponential equations include:

• A base, which is the number being multiplied.
• An exponent, indicating how many times the base is used in multiplication.
• A result, which is the outcome of the base raised to the exponent.

For example, in the equation \( 64^{\frac{1}{3}} = 4 \), 64 is the base, \( \frac{1}{3} \) is the exponent, and 4 is the result. Understanding this structure is crucial when learning to manipulate these equations, especially in converting to other forms such as logarithmic form.

Logarithms

Logarithms are a fundamental concept in mathematics that serve as the inverse operation to exponentiation. While exponentiation raises a number to a power, a logarithm tells us what power a base must be raised to, to reach a specific number.Understanding logs is essential since:

• They simplify complex multiplication and division into addition and subtraction, respectively.
• Logs are used in various applications like solving exponential equations and analyzing data trends.
• They provide a way to express very large or small numbers, common in scientific calculations.

For instance, the logarithmic expression \( \log_{64}(4) \) asks, "To what power must 64 be raised to obtain 4?" In this case, the answer is \( \frac{1}{3} \). Grasping this concept is crucial for converting between exponential and logarithmic forms.

Conversion Between Exponential and Logarithmic Form

Conversion between exponential and logarithmic forms is an important skill that simplifies handling equations and understanding relationships between numbers. The process allows for transitions between a number's exponent and its logarithmic representation.Consider the basic equation form:

• Exponential form: \( b^e = a \)
• Logarithmic form: \( \log_b(a) = e \)

This means, if you know information about two parts of this equation, you can find the third. For example, given \( b = 64 \), \( a = 4 \), and \( e = \frac{1}{3} \), you can fluently switch from the exponential equation \( 64^{\frac{1}{3}} = 4 \) to its logarithmic counterpart \( \log_{64}(4) = \frac{1}{3} \). This conversion is instrumental in solving complex equations in both algebra and calculus, and provides a deeper understanding of the interplay between bases, exponents, and results.