Problem 29 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 27 \).

Step 1: Identify the Logarithmic Equation

The given equation is \( \log_{3}(x) = 3 \). This is a logarithmic equation where the base is 3, the argument is \( x \), and the logarithm is equal to 3.

Step 2: Convert Logarithmic Equation to Exponential Form

To convert the logarithmic equation \( \log_{3}(x) = 3 \) to exponential form, you write it as \( 3^3 = x \). Here, 3 is the base, 3 is the exponent, and \( x \) is the result.

Step 3: Solve the Exponential Equation

With the equation \( 3^3 = x \), calculate \( 3^3 \). This means 3 multiplied by itself 3 times, which equals 27.

Step 4: Verify the Solution

Verify the solution by substituting \( x = 27 \) back into the original logarithmic equation: \( \log_{3}(27) = 3 \). Since \( 27 = 3^3 \), taking the logarithm base 3 of 27 gives 3, confirming the solution is correct.

Key Concepts

Exponential Form

Logarithmic equations can often be more easily managed by rewriting them in exponential form. When we have an equation like \( \log_{b}(a) = c \), this can be transformed into its exponential form, \( b^c = a \).
This is quite valuable because exponential equations are sometimes more straightforward to solve.
The standard pattern involves:

• Base \( b \): This is the number raised to a power. In our example, 3 is the base.
• Exponent \( c \): Indicates how many times the base is multiplied by itself. With \( \log_{3}(x) = 3 \), the exponent is 3.
• Result \( a \): The product obtained after raising the base to the exponent. In our case, it's \( x \), which we solve for.

Understanding this form helps simplify the solving process as we shift the equation from logarithms to the more familiar territory of exponentiation.

Base and Argument

In any logarithmic expression \( \log_{b}(a) = c \), understanding the base and argument is crucial.
The base, denoted as \( b \), is the number from which you "count the powers." In our problem, \( 3 \) is the base, meaning we are considering powers of 3.
The argument, shown as \( a \), is the number we take a logarithm of. It's essential because it is often the value we are interested in finding. In this scenario, the argument is \( x \).
The logarithm itself, which results in \( c \), is what tells us the power to which the base must be raised to equal the argument.
When solving, comprehending these parts helps guide the casting of the logarithmic equation into its exponential equivalent, making these terms less intimidating and more manageable.

Solving Equations

Once a logarithmic equation is converted into exponential form, solving it becomes much clearer.
Consider the converted equation from the example: \( 3^3 = x \). Solving for \( x \) entails evaluating \( 3^3 \), which means multiplying 3 by itself three times:
\( 3 \times 3 \times 3 = 27 \). Thus, \( x = 27 \).
The final key step in problem-solving is verification. This confirms that the result is correct. Substitute the solution back into the original equation. If We have \( \log_{3}(27) = 3 \), verify by realizing that since 27 is \( 3^3 \), the equation holds.
Thus, solving involves simplifying to exponential form, computing the powers, and verifying, ensuring all steps align correctly.