Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{9}(x)=\frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \( x \) is 3.

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{9}(x) = \frac{1}{2} \). This represents a logarithm with base 9 equaling \( \frac{1}{2} \). Our goal is to solve for \( x \).

Step 2: Convert Logarithmic to Exponential Form

The property of logarithms states that if \( \log_{b}(a) = c \), then the equivalent exponential form is \( b^c = a \). Applying this property to the given equation, \( \log_{9}(x) = \frac{1}{2} \) converts to the exponential form \( 9^{\frac{1}{2}} = x \).

Step 3: Simplify the Exponential Expression

The expression \( 9^{\frac{1}{2}} \) represents the square root of 9. Calculating the square root gives \( \sqrt{9} = 3 \). Thus, \( x = 3 \).

Step 4: Final Solution

By converting the logarithm to exponential form and simplifying, we find that \( x = 3 \).

Key Concepts

Logarithmic Equation

A logarithmic equation is essentially a way of expressing an exponent in a more compact form. For example, the equation \( \log_{9}(x) = \frac{1}{2} \) states that the power to which the base \( 9 \) must be raised to obtain \( x \) is \( \frac{1}{2} \). Logarithmic equations are useful because they provide a way to solve equations involving exponentials in a more approachable manner. When you come across a logarithmic equation, your goal is often to find the unknown variable that balances the equation. This typically involves converting the log equation into an exponential form, which can be more straightforward to solve.

Solving for x

Solving for \( x \) in a logarithmic equation involves a series of steps designed to isolate the variable. First, you start by recognizing the form of the equation. Here, our equation is \( \log_{9}(x) = \frac{1}{2} \). The task is to determine what value of \( x \) satisfies this equation. We convert the logarithmic expression into its exponential form. By using the property \( \log_{b}(a) = c \Rightarrow b^c = a \), we convert \( \log_{9}(x) = \frac{1}{2} \) into \( 9^{\frac{1}{2}} = x \). This transformation makes solving for \( x \) much simpler as you can directly compute the expression.

Properties of Logarithms

Logarithms have several key properties that make them indispensable in algebra. Understanding these properties helps in rewriting and solving logarithmic equations. Some basic properties include:

• Product Property: \( \log_b(MN) = \log_b M + \log_b N \)
• Quotient Property: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Property: \( \log_b (M^k) = k\log_b M \)

These properties can help simplify complicated logarithmic expressions and equations, making it easier to convert them into exponential form for solving. For the given problem, the main property at play was understanding how to convert between logarithmic and exponential forms, which is foundational for manipulating these expressions.

Simplifying Exponential Expressions

Simplifying exponential expressions is a vital skill when working with logarithms. Once you've converted a logarithmic equation into an exponential form, as we did with \( 9^{\frac{1}{2}} = x \), your task becomes simplifying this expression. The superscript \( \frac{1}{2} \) implies taking the square root. Therefore, \( 9^{\frac{1}{2}} \) is equivalent to finding \( \sqrt{9} \), which simplifies to \( 3 \). As a result, in this scenario, \( x = 3 \). This simplification process is straightforward once you understand the relationship between exponents and roots, allowing you to tackle similar problems with confidence.