Problem 153 - Precalculus

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

\( x = 3 \)

Step 1: Understand Logarithmic Equation

The given equation is \( \log_{9}(x) = \frac{1}{2} \). This represents that when the base 9 is raised to the power of \( \frac{1}{2} \), it equals \( x \).

Step 2: Convert to Exponential Form

Convert the logarithmic equation to its exponential form. The equation \( \log_{b}(y) = z \) can be rewritten as \( b^z = y \). Thus, \( \log_{9}(x) = \frac{1}{2} \) can be rewritten as:\[ x = 9^{\frac{1}{2}} \]

Step 3: Simplify the Exponential Expression

The expression \( 9^{\frac{1}{2}} \) represents the square root of 9. Simplify this expression:\[ x = \sqrt{9} \] which equals 3.

Key Concepts

Exponential Form

Understanding the exponential form is key when dealing with logarithmic equations. Exponential form takes the expression from a logarithm and transforms it. When you see an equation in the form of \( \log_{b}(y) = z \), you can rewrite it as \( b^z = y \). Here:

• \( b \) is the base of the logarithm.
• \( y \) is the result or the argument of the logarithm.
• \( z \) is the exponent or the value the base is raised to.

Using the exponential form makes it easier to visualize and solve logarithmic equations. In our exercise, the conversion of \( \log_{9}(x) = \frac{1}{2} \) to \( x = 9^{\frac{1}{2}} \) helps simplify the problem by expressing \( x \) as a result of 9 raised to the power of \( \frac{1}{2} \), providing clarity in solving for \( x \).

Logarithms

Logarithms are a fundamental concept in mathematics. They provide a method to express exponential relationships, but in reverse. Simply put, a logarithm answers the question: "To what power must we raise a given base to achieve a specific number?" For example, in the expression \( \log_{9}(x) = \frac{1}{2} \), we are asking what power \( 9 \) must be raised to in order to equal \( x \).Key characteristics of logarithms include:

• They transform multiplicative relationships into additive ones.
• They have a base, which determines the number we're repeatedly multiplying.
• Their output, or logarithmic value, is the exponent needed to achieve the target value.

Understanding logarithms can significantly simplify the process of solving equations where exponential growth or decay is involved.

Solving Equations

Solving equations involves finding the value of an unknown variable that makes the equation true. In the case of logarithmic equations, converting to exponential form is a strategic step. It often simplifies the equation and makes it more approachable.Here’s a brief guide to solving logarithmic equations:

• Identify the logarithmic form in your equation, such as \( \log_{b}(y) = z \).
• Convert it to exponential form: \( b^z = y \).
• Simplify the equation: evaluate the exponential operation if possible.
• Solve for the unidentified variable—like \( x \) in our example.

In our exercise, once the equation was converted to \( x = 9^{\frac{1}{2}} \), further simplifying it by evaluating the exponential expression gives \( x = 3 \). This systematic approach demystifies the process of leveraging logarithmic and exponential relationships to solve mathematical problems.