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Chapter 9: Problem 86

Solve $$ \log _{x} 9=\frac{1}{2} $$

Short Answer

Expert verified
x = 81

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\log_{x} 9 = \frac{1}{2}\). This means that x raised to the power of \( \frac{1}{2} \) equals 9.
02

Rewrite the equation in exponential form

To solve for x, convert the logarithmic equation to its exponential form: \( x^{\frac{1}{2}} = 9 \).
03

Square both sides of the equation

Square both sides to eliminate the exponent \( \frac{1}{2} \): \((x^{\frac{1}{2}})^{2} = 9^{2}\). This simplifies to \( x = 81 \).
04

Verify the solution

Substitute x back into the original equation to verify: \( \log_{81} 9 = \frac{1}{2} \). Indeed, \( 81^{\frac{1}{2}} = 9\), confirming that x = 81 is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential form
To solve logarithmic equations, converting to exponential form is essential. The exponential form of a logarithm helps us rewrite the equation in a way that is easier to solve.

Let鈥檚 start with the given equation: \( \log_{x} 9 = \frac{1}{2} \).
This means that we need to find x such that when raised to the power of \frac{1}{2}, it equals 9. We can rewrite our equation in exponential form. Instead of having it in the format of log, we aim for:
\( x^{\frac{1}{2}} = 9 \)

This transformation allows us to directly apply algebraic methods to solve for x. Understanding this step helps bridge logarithms with exponents and offers a clearer path to follow.
solving logarithms
After converting to the exponential form, the next task is to solve the equation. Here, we have:
\( x^{\frac{1}{2}} = 9 \)

To eliminate the fraction as an exponent, square both sides of the equation.

Step 1: Square both sides:
\((x^{\frac{1}{2}})^{2} = 9^{2}\)

Now it becomes:
\ x = 81 \

This step effectively removes the fraction exponent by raising both sides to a power that cancels it out. It's crucial to remember this technique when dealing with fractional exponents.
verification of solutions
After finding the solution, always verify its correctness by substituting back into the original equation.

For our example, substitute x back in the original logarithmic equation:
\ \log_{81} 9 = \frac{1}{2} \

Remember that:- 81 raised to the \(\frac{1}{2}\) power should equal 9.

By calculating, we confirm:
81^{\frac{1}{2}} = 9

This demonstrates that our solution x = 81 is indeed accurate. Verification is crucial to ensure that the derived values are correct and meet the equation's conditions.

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