Problem 71 - Intermediate Algebra

Question

Solve for \(x .\) See Example 3. $$ \log _{8} x=0 $$

Step-by-Step Solution

Answer

The value of \( x \) is 1.

Step 1: Understand the Problem

We are given the equation \( \log_{8}x = 0 \). The goal is to find the value of \( x \) that satisfies this equation. The expression \( \log_{8}x \) means 'the power to which we must raise 8 to get \( x \)'.

Step 2: Apply the Definition of Logarithm

The equation \( \log_{8}x = 0 \) can be rewritten using the definition of logarithms. This means that 8 raised to the power of 0 gives us \( x \). So, we have:\[ x = 8^0 \]

Step 3: Evaluate the Expression

Calculate \( 8^0 \). Any non-zero number raised to the power of 0 is 1:\[ 8^0 = 1 \]

Step 4: Conclusion

Since \( x = 8^0 \) and \( 8^0 = 1 \), it follows that:\[ x = 1 \]

Key Concepts

Definition of Logarithm

A logarithm is a mathematical concept that answers the question: "To what power must the base, typically a number like 10 or 2, be raised to produce a given number?" For instance, in the expression \( \log_{b} a \), \( b \) is the base, and \( a \) is the result of raising \( b \) to some power. The logarithm answers the query: " What power should \( b \) be raised to get \( a \)?" This can be more simply put as:

• \( \log_{b} a = c \) means that \( b^c = a \).
• Example: \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).

Logarithms are a way to reverse the process of exponentiation, much like subtraction undoes addition. In the exercise we chose, \( \log_8 x = 0 \), we're being asked what power we need to raise 8 in order to get \( x \). This forms the basis of solving logarithmic equations.

Exponent Properties

Exponents are powerful ways to represent repeated multiplication. They follow specific, useful rules that simplify handling mathematical problems.One key property, known as the "Zero Exponent Rule," states that any non-zero number raised to the power of zero is 1. This can be mathematically represented as:

• \( a^0 = 1 \) for any \( a eq 0 \).
• Example: \( 8^0 = 1 \).

In the example exercise, we used this property to solve for \( x \). Since \( 8^0 = 1 \), the equation \( x = 8^0 \) simplifies directly to \( x = 1 \). Understanding exponent properties like these makes solving equations much simpler. They give us the ability to express and manipulate large numbers with ease and precision.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value of unknown quantities that satisfy given conditions. Here, we focused on logarithmic equations, where logarithms and their properties help to unwind the relationship between numbers.Let's break down the steps typically involved in solving such problems:

• Identify the Equation: Clearly understand what the equation is asking. For example, \( \log_8 x = 0 \) asks for the value of \( x \) that results when 8 is raised to the power of 0.
• Rewrite Using Definitions: Use your understanding of the definition of logarithms to rewrite the equation. In our example, we turn it into the equivalent form: \( x = 8^0 \).
• Evaluate: Apply necessary mathematical rules and properties, like evaluating \( 8^0 \), which results in 1.
• State the Conclusion: After calculation, conclude with a clear solution: here, \( x = 1 \).

Through practice, these steps become more intuitive, allowing you to tackle logarithmic equations with confidence. Each problem will further solidify your understanding of how these mathematical concepts interconnect.