Problem 52 - College Algebra

Question

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. $$ \log _{3}(x)+3=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{3} \).

Step 1: Isolate the Logarithmic Function

To solve the equation \( \log_{3}(x) + 3 = 2 \), start by isolating the logarithmic function. Subtract 3 from both sides of the equation: \[ \log_{3}(x) = 2 - 3 \] which simplifies to \[ \log_{3}(x) = -1 \].

Step 2: Convert from Logarithmic to Exponential Form

The equation \( \log_{3}(x) = -1 \) can be rewritten in exponential form to eliminate the logarithm. Recall that \( \log_{b}(a) = c \) is equivalent to \( b^{c} = a \). Thus, \( 3^{-1} = x \).

Step 3: Simplify the Exponential Expression

Now, calculate \( 3^{-1} \): \[ x = \frac{1}{3} \]. Therefore, the solution to the equation is \( x = \frac{1}{3} \).

Step 4: Graph to Verify Solution

Graph the functions \( y = \log_{3}(x) + 3 \) and \( y = 2 \) on the same set of axes. Observe the point of intersection of these graphs. The intersection occurs where \( x = \frac{1}{3} \), verifying the solution. Since the logarithm is only defined for positive \( x \), ensure the function and graph reflect this.

Key Concepts

Exponential Form

Understanding exponential form is key to solving logarithmic equations efficiently. Logarithmic and exponential forms are closely linked. When we have a logarithmic equation like \( \log_{3}(x) = -1 \), we can convert it to exponential form to simplify our calculations.

To convert, remember that if \( \log_{b}(a) = c \), it implies \( b^{c} = a \). Applying this conversion, \( \log_{3}(x) = -1 \) becomes \( 3^{-1} = x \).

• This covariance of logarithmic and exponential forms helps greatly in equation solving.
• Exponential form gives a direct method to find \( x \) without logarithmic complications.

With this method, we quickly find \( x = \frac{1}{3} \). Exponential form is indeed a powerful way to grasp the solution straightforwardly.

Graphing Solutions

Graphing can be an insightful tool to verify solutions and understand equations better. After solving the equation \( \log_{3}(x) + 3 = 2 \), use graphing to visually comprehend the solution.

Plot the function \( y = \log_{3}(x) + 3 \) and the line \( y = 2 \) on the same graph.

• The intersection point tells us where the equality holds, which is our solution.
• Here, the intersection occurs at \( x = \frac{1}{3} \), confirming the solution we calculated analytically.

Graphing the solution helps in not just verifying it, but also understanding the behavior of logarithmic functions in relation to linear functions.

Logarithmic Function

Logarithmic functions have unique properties making them interesting to study. In our equation \( \log_{3}(x) + 3 = 2 \), isolating the logarithmic component is crucial before further manipulation.

Once isolated, you deal with \( \log_{3}(x) = -1 \).

• Logarithms answer the question of what power a base must be raised to get a certain number.
• In this case, what power must 3 be raised to, to yield \( \frac{1}{3} \)?

The equation above precisely manages this relationship, indicating the elegance and utility of logarithmic functions in mathematical formulations.

Equation Solving

Equation solving is a systematic process of finding unknowns and verifying results. In this exercise, solving \( \log_{3}(x) + 3 = 2 \) involves several clear steps.

You start by isolating the logarithmic portion to derive \( \log_{3}(x) = -1 \).

• Next, convert to exponential form: this makes \( x \) explicit as \( 3^{-1} \).
• The computations simplify to find \( x = \frac{1}{3} \).
• Finally, verify by graphing to confirm both sides meet at this value.

Each stage ensures accuracy and confidence in the solution, highlighting the methodical nature of solving equations.