Problem 26 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \log _{3} \sqrt{x^{2}+17}=2 $$

Step-by-Step Solution

Answer

The solutions are \( x = 8 \) and \( x = -8 \).

Step 1: Convert the Logarithmic Equation to Exponential Form

Begin by recognizing that the logarithmic form \( \log_{b}(A) = C \) is equivalent to the exponential form \( A = b^C \). Here, the equation is \( \log_{3} \sqrt{x^{2}+17} = 2 \). Thus, we write it in exponential form as: \[ \sqrt{x^2 + 17} = 3^2 \].

Step 2: Simplify the Exponential Equation

Calculate \( 3^2 \), which equals 9. Replace this in the equation, resulting in: \[ \sqrt{x^2 + 17} = 9 \].

Step 3: Eliminate the Square Root

To remove the square root, square both sides of the equation. This gives: \((\sqrt{x^2 + 17})^2 = 9^2 \) which simplifies to \( x^2 + 17 = 81 \).

Step 4: Solve for x

Subtract 17 from both sides to isolate \( x^2 \). This results in: \[ x^2 = 81 - 17 \], which simplifies to \( x^2 = 64 \).

Step 5: Take the Square Root

To find \( x \), take the square root of both sides of the equation. Remember that taking the square root could yield a positive or negative result: \( x = \pm \sqrt{64} \). Since \( \sqrt{64} = 8 \), we have \( x = 8 \) and \( x = -8 \).

Step 6: Verify the Solutions

Verify both potential solutions. Substitute \( x = 8 \) into the original equation: \( \log_{3} \sqrt{8^2 + 17} \). Calculate \( 64 + 17 = 81 \), and \( \sqrt{81} = 9 \), so \( \log_{3} 9 = 2 \), which is correct. Similarly, \( x = -8 \) results in the same validity, as the computation within the square root becomes \( \sqrt{(-8)^2 + 17} = 9 \).

Key Concepts

Exponential Form

When working with logarithmic equations, it's crucial to understand the transition from logarithmic to exponential form. This conversion involves rewriting an equation from the logarithmic form \( \log_b(A) = C \) into its equivalent exponential form \( A = b^C \). By doing this, we can often simplify the problem significantly, making it easier to find the solution.In our exercise, we have the logarithmic equation \( \log_{3} \sqrt{x^{2}+17} = 2 \). To convert this to exponential form, we recognize that it's asking what power \( 3 \) must be raised to, in order to result in \( \sqrt{x^{2}+17} \). This yields the equation \( \sqrt{x^{2}+17} = 3^2 \). Converting to exponential form is a powerful strategy to simplify equations when solving for unknown values.

Square Root Elimination

Once we have the equation in an exponential form, we often encounter expressions that make it necessary to eliminate a square root. This is done by squaring both sides of the equation, effectively removing the square root symbol and allowing us to solve for the variable inside.For our specific equation, \( \sqrt{x^2 + 17} = 9 \), we make the next logical step by squaring both sides. This action results in \( (\sqrt{x^2 + 17})^2 = 9^2 \), simplifying to \( x^2 + 17 = 81 \). This crucial step transforms a potentially tricky radical expression into a straightforward algebraic equation, setting the stage for easy solution of \( x \) in subsequent steps.

Solution Verification

After finding potential solutions to an equation, it is always wise to check their validity. Verification not only ensures we have the correct solution, but it can also reveal any extraneous results introduced during mathematical manipulations such as squaring.When verifying our solutions \( x = 8 \) and \( x = -8 \) for the equation \( \log_{3} \sqrt{x^{2}+17} = 2 \), we substitute both back into the original equation. For \( x = 8 \), the expression becomes \( \log_{3} \sqrt{8^2 + 17} = \log_{3} 9 \), which simplifies to 2, confirming that it holds true. Similarly for \( x = -8 \), the expression simplifies in the same manner, also confirming its validity. By checking each solution, we verify that our approach is correct and that these values satisfy the original logarithmic equation.