Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log _{324}(18)=\frac{1}{2}\)

Step-by-Step Solution

Answer

\( 324^{\frac{1}{2}} = 18 \).

Step 1: Identify the Logarithmic Form

We begin with the logarithmic equation provided: \( \log_{324}(18) = \frac{1}{2} \). This is in the form \( \log_{b}(x) = y \).

Step 2: Convert Logarithmic to Exponential Form

Using the relationship \( \log_{b}(x) = y \) is equivalent to \( b^y = x \), we rewrite the equation \( \log_{324}(18) = \frac{1}{2} \) as an exponential equation. Therefore, \( 324^{\frac{1}{2}} = 18 \).

Step 3: Verify the Conversion

To verify, calculate \( 324^{\frac{1}{2}} \), which equals \( \sqrt{324} \). The square root of 324 is indeed 18, affirming that our conversion is correct.

Key Concepts

Exponential Equations

Exponential equations play a key role in mathematics and represent a distinct way of expressing growth or decay. They typically look like this: \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result. In simple terms, we raise a number (the base) to a power (the exponent) to get another number (the result).

For instance, in the equation \( 324^{\frac{1}{2}} = 18 \), \( 324 \) is the base, \( 1/2 \) is the exponent, and \( 18 \) is the result. In this case, the number 324 is raised to the power of 1/2, which is equivalent to the square root of 324, yielding 18. Remember, any number raised to the power of 1/2 is like asking for its square root.

Understanding exponential equations helps us solve many real-world problems, such as calculating compound interest, population growth, and radioactive decay. These equations are foundational in advanced areas of study, such as calculus and complex systems.

Logarithmic Form

The logarithmic form is another way to express the relationship between numbers through an exponent. It takes the general form \( \log_b(x) = y \). This is read as "the logarithm of \( x \) with base \( b \) is \( y \)." In essence, this asks what power you must raise \( b \) to, in order to obtain \( x \).

To illustrate, look at the equation \( \log_{324}(18) = \frac{1}{2} \). This means you need to raise 324 to the power of 1/2 to get 18.

Logarithms simplify multiplication and division into addition and subtraction, easing calculations greatly—and aiding in solving exponential equations. They appear everywhere, from computer science to earthquake measurement scales, making them a valuable concept to understand.

Conversion Verification

Conversion verification is a crucial step in mathematics to ensure that transformations between different forms have been done correctly. After converting a logarithmic form to an exponential form, checking ensures no calculation errors.

For example, when we converted \( \log_{324}(18) = \frac{1}{2} \) to its exponential form \( 324^{\frac{1}{2}} = 18 \), verification requires evaluating \( 324^{\frac{1}{2}} \). This calculation, revealing \( \sqrt{324} \), equals 18, confirming the accuracy.

Verifying conversions is essential because it spotchecks mathematical work and confirms that subsequent calculations based on these transformations will be trustworthy. This practice cultivates meticulous attention to detail, a significant asset in both academic contexts and everyday problem-solving scenarios.