Problem 33 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ x=\log _{8} 64 $$

Step-by-Step Solution

Answer

\( 8^x = 64 \) is the exponential form of the equation.

Step 1: Understand the logarithmic equation

The given logarithmic equation is \( x = \log_{8} 64 \). This means that the base of the logarithm is 8, the argument is 64, and \( x \) is the result.

Step 2: Set up the equivalent exponential equation

The general form of a logarithmic equation \( y = \log_{b} a \) can be rewritten in exponential form as \( b^y = a \). In our specific problem, this translates to \( 8^x = 64 \).

Step 3: Write the final exponential equation

Based on Step 2, rewrite the given logarithmic equation \( x = \log_{8} 64 \) as \( 8^x = 64 \). This is the equivalent exponential equation.

Key Concepts

Logarithmic Equations

Logarithmic equations involve the concept of logarithms, which are closely related to exponents. In a logarithmic equation, an expression is set equal to a logarithm, which indicates how many times you need to multiply the base number to obtain another number. Understanding logarithmic equations is fundamental in mathematics since they help solve problems involving exponential growth, such as in finance or science scenarios.

For example, in the equation \( x = \log_{8} 64 \), the base is 8. This equation is basically asking: "To what power must 8 be raised, to result in 64?" The answer to this question, \( x \), is the solution to the equation. Here, thinking about logarithms as the inverse function of exponents is key to solving problems that involve finding the unknown raised to a certain power.

Converting Logarithmic to Exponential Form

Converting logarithmic equations to exponential form is a crucial skill that simplifies solving them. The process involves understanding that a logarithm is essentially asking for an exponent. To convert a logarithm to its exponential form, you'll need to rearrange the components of the logarithmic equation.

Let's use our example \( x = \log_{8} 64 \). We need to rearrange this into the form \( b^y = a \). The base \( b \) is 8, the result \( y \) is \( x \), and the argument \( a \) is 64. By rewriting, we get \( 8^x = 64 \). This exponential equation reverses the logarithmic operation, making it easier to grasp and solve using exponent rules.

Here is a quick tip to remember: Whatever the base of the logarithm, that same number becomes the base of the exponential equation. The result of the logarithmic equation becomes the exponent in the exponential equation.

Exponential Form of Logarithms

The exponential form of logarithms expresses a logarithmic equation in terms of exponents and powers. It represents the fundamental relationship between logarithms and powers, making it easier to understand logarithmic functions by showing how they relate to simple exponentiation.

• The base of the logarithm becomes the base of the exponent.
• The outcome of the logarithm acts as the exponent in the converted equation.
• The argument of the logarithm equals the result of the power function.

For \( x = \log_{8} 64 \), translating this into an exponential equation yields \( 8^x = 64 \). This conversion gives us a new perspective. It tells us that multiplying 8 by itself some number of times (specifically, \( x \) times) yields 64.

Understanding this relationship is crucial, especially when you deal with solving equations involving unknown exponents where you are not directly dealing with logarithms, but realizing implicitly how they communicate exponent SHAPES and calculations.