Problem 78 - Intermediate Algebra

Question

Find each value of \(x .\) $$ \log _{2} \sqrt{2}=x $$

Step-by-Step Solution

Answer

The value of \( x \) is \( \frac{1}{2} \).

Step 1: Understand the Given Equation

You are given the logarithmic equation \( \log_{2} \sqrt{2} = x \). This equation means: 'To what power must 2 be raised, in order to get \( \sqrt{2} \)?' Our task is to find this power, i.e., \( x \).

Step 2: Rewrite the Square Root as an Exponent

Rewrite \( \sqrt{2} \) using exponents. Recall that the square root of a number \( a \) is the same as \( a \) raised to the power of \( \frac{1}{2} \). Thus, \( \sqrt{2} = 2^{\frac{1}{2}} \).

Step 3: Apply the Logarithm Power Rule

The logarithm power rule states that \( \log_{b}(a^n) = n \cdot \log_{b}(a) \). Apply this rule to the expression, substituting for \( \sqrt{2} \):\[\log_2(2^{\frac{1}{2}}) = \frac{1}{2} \cdot \log_{2}(2)\]

Step 4: Solve for \( x \) Using the Definition of Logarithms

Since \( \log_{2}(2) = 1 \) (because 2 raised to 1 equals 2), we substitute this back into the equation:\[\frac{1}{2} \cdot \log_{2}(2) = \frac{1}{2} \cdot 1 = \frac{1}{2}\]Thus, \( x = \frac{1}{2} \).

Key Concepts

Understanding Exponential Expressions

In mathematics, an exponential expression refers to the representation of numbers where a base number is raised to a variable or constant power. For example, the expression \(2^{x}\) means the number 2 is raised to the power of \(x\). This notation is particularly useful in logarithmic and exponential equations.
In the exercise, we handled \(\sqrt{2}\) by rewriting it as \(2^{\frac{1}{2}}\). This step is crucial because it allows us to express a square root using exponential notation, which simplifies further calculations.
If you're dealing with other roots, remember these key points:

• The square root of a number \(a\) can be rewritten as \(a^{\frac{1}{2}}\).
• The cube root becomes \(a^{\frac{1}{3}}\).
• In general, the \(n\)-th root of a number \(a\) is \(a^{\frac{1}{n}}\).

Recognizing these forms helps when transitioning between exponential and logarithmic forms.

Applying the Logarithm Power Rule

The logarithm power rule is a handy tool that simplifies the process of solving logarithms with powers. It states that \(\log_{b}(a^n) = n \cdot \log_{b}(a)\). This rule essentially allows you to "pull the exponent out" of the logarithm by placing it in front as a coefficient.
In our example, once we rewrote \(\sqrt{2}\) as \(2^{\frac{1}{2}}\), we applied this rule:

• We took \(\log_2(2^{\frac{1}{2}})\).
• Utilizing the power rule, we knew it could be expressed as \(\frac{1}{2} \cdot \log_{2}(2)\).

This transformation shows how exponents interact with logarithms, making such logarithmic calculations much more manageable.Understanding this rule is important as it transforms complex logarithmic expressions and often simplifies the process of solving equations involving logarithms and exponents.

Solving for Variables in Logarithmic Equations

To solve for variables in logarithmic equations, you need a strong grasp of how logarithms and exponents work together. The end goal is often to isolate the variable of interest.In our example, we had to determine the value of \(x\) in \(\log_{2} \sqrt{2} = x\). Here's how we solved it:

• First, we converted \(\sqrt{2}\) into \(2^{\frac{1}{2}}\).
• Then, using the logarithm power rule, we wrote \(\log_{2}(2^{\frac{1}{2}})\) as \(\frac{1}{2} \cdot \log_{2}(2)\).
• Knowing that \(\log_{2}(2) = 1\), we simplified this to \(\frac{1}{2} \cdot 1 = \frac{1}{2}\).

As a result, we found that \(x = \frac{1}{2}\).In many logarithmic equations, identifying similarities and applying fundamental rules can swiftly lead you to the solution. The key steps involve transforming the equation's form, applying known logarithmic rules, and methodically working through arithmetic until the variable is isolated.