Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{3}(x)=2\)

Step-by-Step Solution

Answer

The solution is \( x = 9 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{3}(x) = 2 \). This means that the logarithm with base 3 gives us the result 2 when applied to \(x\).

Step 2: Convert to Exponential Form

To convert the logarithmic equation \( \log_{3}(x) = 2 \) to exponential form, we use the definition of a logarithm: if \( \log_{b}(a) = c \), then \( b^{c} = a \). So, we rewrite the equation as \( 3^{2} = x \).

Step 3: Calculate the Exponential Result

Now, compute the value of \( 3^{2} \) to find \(x\). Since \( 3^{2} = 9 \), it follows that \( x = 9 \).

Step 4: Verify the Solution

To ensure the solution is correct, substitute \( x = 9 \) back into the original logarithmic equation: \( \log_{3}(9) \). Since \( 3^{2} = 9 \), this confirms that \( \log_{3}(9) = 2 \). Hence, the solution is valid.

Key Concepts

Exponential Form

Understanding how to convert logarithmic equations into exponential form is a vital skill when solving these types of problems. At the heart of the process lies the fundamental definition of a logarithm. The logarithm answers the question: "To what power should the base be raised, to produce a given number?" If you have a logarithmic equation like \( \log_{b}(a) = c \), converting it into an exponential equation involves rearranging it to the form \( b^{c} = a \). This transformation is crucial as it allows the solving of equations that might initially seem complex. Converting the equation \( \log_{3}(x) = 2 \) to exponential form makes it easier to see that the equation can also be expressed as \( 3^{2} = x \). This step essentially tells us that the number 3, raised to the power of 2, will result in \( x \). It's a straightforward yet powerful method to simplify and solve logarithmic equations.

Logarithms

Logarithms are a mathematical concept that helps us determine the power to which a number, known as the base, must be raised to obtain another number. This is particularly useful in scenarios involving exponential growth. When you encounter \( \log_{3}(x) = 2 \), the notation implies that you are trying to find which number 3 needs to be raised to, in order to get \( x \). In this case, it is clear that \( 3^2 = 9 \), so \( x \) must be 9. Logarithms were initially devised to simplify calculations and were revolutionary in speeding up mathematical computations. Key properties to remember about logarithms are:

• The logarithm of a product is the sum of the logarithms: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \).
• The logarithm of a quotient is the difference of the logarithms: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \).
• The logarithm of a power is the exponent times the logarithm: \( \log_{b}(m^{n}) = n \times \log_{b}(m) \).

These properties make logarithms a powerful tool in various fields including science, engineering, and computer science.

Step-by-Step Solution

Solving logarithmic equations step-by-step is often the best way to reach the correct answer, especially if you're new to the concept. Let's break down the process using an example of \( \log_{3}(x) = 2 \).

• Step 1: Interpret the logarithmic equation. Recognize that \( \log_{3}(x) \equiv 2 \) means that 3, taken to the power of 2, equals \( x \).
• Step 2: Convert to exponential form. Using the definition of a logarithm, rewrite it as \( 3^{2} = x \). This transformation reveals the relationship between the numbers involved.
• Step 3: Compute the exponential result. Calculate \( 3^{2} \), which equals 9. Hence, \( x = 9 \).
• Step 4: Verify the solution. Substitute \( x \) back into the original equation to ensure it's correct: \( \log_{3}(9) \equiv 2 \). Since \( 3^{2} \) does indeed equal 9, the solution checks out.

Following these steps carefully not only helps in arriving at the right solution but also strengthens your understanding of how to tackle logarithmic problems effectively. Step-by-step practice can turn difficult-looking equations into clear and easy-to-solve problems.