Problem 26 - College Algebra

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 a logarithmic equation of the form \( \log_b(a) = c \). In this case, \( b = 3 \), \( a = x \), and \( c = 2 \).

Step 2: Convert Logarithmic to Exponential Form

To convert a logarithmic equation to its exponential form, use the rule \( \log_b(a) = c \Leftrightarrow b^c = a \). Here, substitute the values: \( 3^2 = x \).

Step 3: Calculate the Exponential Expression

Compute \( 3^2 \). Since \( 3^2 = 3 \times 3 \), it simplifies to \( 9 \).

Step 4: Solution for \( x \)

From the exponential equation \( x = 9 \), we conclude that the solution to the logarithmic equation is \( x = 9 \).

Key Concepts

Exponential Form Conversion

To understand how to convert a logarithmic equation into exponential form, it's essential to remember one key rule: if you have a logarithmic equation \( \log_b(a) = c \), it is equivalent to the exponential equation \( b^c = a \). This conversion is useful as it transforms a logarithmic problem, which might seem complex, into an exponential problem that is often simpler to solve.
Breaking it down:

• The base \( b \) in the logarithm becomes the base in the exponent.
• The logarithm result \( c \) becomes the exponent.
• The number \( a \) inside the logarithm becomes the result of the exponentiation.

In our example, \( \log_3(x) = 2 \) converts to \( 3^2 = x \). This step is like translating a sentence in one language into another, retaining the same meaning but in a more understandable form.

Logarithmic Equations

Logarithmic equations involve logarithms, which are a way to find out how many times you multiply a base by itself to get another number. In our problem, \( \log_3(x) = 2 \), the equation asks, "what power must we raise 3 to obtain x?"
Logarithmic equations can be solved by converting them into exponential form, making them more approachable. The key element here is understanding how the terms in a logarithm interact. The base affects both the size and the direction when you manipulate these numbers.
Consider these simple points:

• The base of the logarithm determines the multiplication factor.
• The exponent tells you how many times to multiply that base.
• Solving logarithmic equations often reveals the result as a straightforward number.

Turning logarithmic equations into something we can compute means recognizing the pivotal relationship between logarithms and exponents.

Exponentiation Process

The exponentiation process involves taking a base number and raising it to a specific power. This power indicates how many times you multiply the base by itself. For instance, in the exponential equation \( 3^2 = x \), the action is to raise 3 to the power of 2.
Here's how it plays out:

• Begin with the base (3).
• Multiply the base by itself as many times as indicated by the exponent (2 here).
• Compute the result: \( 3 \times 3 = 9 \).

The result is straightforward—\( x = 9 \), thus concluding that 3 raised to the power of 2 equals 9.
Exponentiation is a fundamental operation in mathematics that allows us to express large numbers concisely and is essential for solving logarithmic equations by converting them to this form.