Question

Find the value of \(x\) in each expression. $$\log _{36} x=\frac{1}{2}$$

Short answer

The value of \(x\) is 6.

Step-by-step solution

Understand the Logarithmic Equation

The given logarithmic equation \(\log _{36} x = \frac{1}{2}\) states that 36 raised to the power of 1/2 is equal to x.

Convert the Logarithmic Form to Exponential Form

To find the value of x, we rewrite the logarithmic equation in exponential form: \(36^{\frac{1}{2}} = x\).

Calculate the Exponential Expression

Since \(36^{\frac{1}{2}}\) is equivalent to the square root of 36, we find that \(\sqrt{36} = 6\), which means \(x = 6\).

Key concepts

Exponential Form Conversion

When working with logarithmic equations, understanding how to convert them into their exponential form is essential. Logarithms are the inverse operations of exponents. For example, the equation \(\log _{b} x = y\) can be rewritten as \(b^{y} = x\). This is what we see in our exercise \(\log _{36} x = \frac{1}{2}\), which is equivalent to \(36^{\frac{1}{2}} = x\).

This step is crucial because it translates the equation into a more familiar form, allowing us to apply our understanding of exponents to find the solution. Students often find this conversion challenging, but keeping the rule 'the base raised to the logarithm of the number equals the number itself' in mind will help simplify any such problems.

Solving Exponential Equations

Solving exponential equations often involves finding the value of the variable that will make the equation true. It's about dealing with expressions that have exponents. In the given exercise, once we have converted to an exponential form, we obtain the expression \(36^{\frac{1}{2}}\).

To solve for \(x\), we can then calculate the value of \(36^{\frac{1}{2}}\) directly or simplify the expression using the properties of exponents. Understanding the relationship between exponents and radicals is crucial here, as demonstrated in Step 3 of the solution. By recognizing that raising a number to the one-half power is the same as taking the square root, we simplify the solution process. Highlighting these connections can further solidify the student's understanding and improve their approach to similar problems.

Radicals and Exponents

Radicals are an alternate way of expressing exponents, particularly fractional exponents. The nth root of a number is equivalent to that number raised to the power of \(1/n\). This is pivotal in solving exponential equations where fractional exponents are involved. For instance, \(36^{\frac{1}{2}}\) is the same as \(\sqrt{36}\).

Grasping this concept allows students to solve exponential equations without necessarily needing a calculator for all the steps. In our exercise, acknowledging the square root of 36 is simply 6 makes the process more intuitive. Providing exercises that further explore the interplay between radicals and exponents can improve students' comfort with these types of algebraic manipulations.