Problem 83 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(\log _{5} \sqrt{x+4}=2\)

Step-by-Step Solution

Answer

x = 621

Step 1: Understand the Given Equation

The equation given is \(\log_{5} \sqrt{x+4} = 2 \). This means taking the logarithm base 5 of the square root of \(x + 4\) results in 2.

Step 2: Rewrite the Logarithmic Equation in Exponential Form

Convert the logarithmic equation to its exponential form. The equation \(\log_{5} y = 2\) can be written as \(y = 5^2\). Therefore, \(\sqrt{x + 4} = 5^2 \).

Step 3: Simplify the Exponential Equation

Calculate \(5^2\) to get 25. Therefore, \(\sqrt{x + 4} = 25\).

Step 4: Square Both Sides

Remove the square root by squaring both sides of the equation: \( (\sqrt{x + 4})^2 = 25^2 \Longrightarrow x + 4 = 625 \).

Step 5: Solve for x

Isolate \(x\) by subtracting 4 from both sides: \( x = 625 - 4 \). This simplifies to \( x = 621 \).

Key Concepts

Logarithms

Logarithms are mathematical operations that help us work with large numbers more easily. A logarithm answers the question, “What power must a base be raised to in order to get a certain number?”

In the equation \(\log_{5} \sqrt{x+4} = 2\), the base is 5. This tells us that 5 raised to a certain power gives the square root of \(x+4\).

Key points to understand about logarithms include:

• Base and exponent relationship: \(\log_{b}(a) = c\) means \(b^c = a\).
• Logarithms are inverse operations of exponentiation.
• Common bases include 10, 2, and Euler’s number (e).

Exponential Form

Exponential form is a way of expressing numbers or expressions using a base raised to a power. Converting logarithmic equations to exponential form is a common technique to simplify solving.

For the given problem, converting \(\log_{5} \sqrt{x+4} = 2\) to exponential form means rewriting it as \(\sqrt{x+4} = 5^2\). This states that 5 raised to the power of 2 equals the square root of \((x+4) \).

Things to remember about exponential form:

• It is often easier to work with than logarithmic form.
• Ensure accuracy in converting between forms.
• Exponential form is essential for solving many logarithmic problems.

Solving Equations

Solving equations is the process of finding the values of variables that make a mathematical statement true. For logarithmic equations, solving often requires converting to exponential form and then isolating the variable.

For \(\log_{5} \sqrt{x+4} = 2\), after converting to \(\sqrt{x+4} = 25\), we square both sides to remove the square root, yielding \(x+4 = 625\). Then, isolating \(x\) by subtracting 4 achieves the solution.

Steps to solve such problems:

• Convert to exponential form if necessary.
• Simplify the equation as much as possible.
• Use algebraic techniques to isolate the variable.

Mathematical Problem Solving

Mathematical problem solving involves critically analyzing and manipulating equations to find solutions. Understanding the steps and methods, like those used in solving logarithmic equations, is crucial.

In our example, the problem requires:

• Understanding the logarithmic equation and its components.
• Converting to exponential form to simplify.
• Applying algebraic methods to find \(x\).

Effective problem-solving strategies include:

• Breaking down the problem into manageable steps.
• Checking each step for accuracy.
• Applying mathematical principles and rules consistently.