Problem 73 - Precalculus with Calculus Previews

Question

Find the points on the graph of the given function that have the indicated \(y\) -coordinate. $$ f(x)=\log _{3}(x+2) ; 2 $$

Step-by-Step Solution

Answer

The point on the graph is (7, 2).

Step 1: Set Up the Equation

We know that we need to find the points on the graph of the function where the y-coordinate is 2. So, we set up the equation using the given function: \[ f(x) = \log_{3}(x+2) = 2 \] where the left side is the function and the right side is the given y-coordinate.

Step 2: Convert the Logarithm to an Exponent

To solve the equation \( \log_{3}(x+2) = 2 \), we convert it to its exponential form: \[ 3^2 = x + 2 \] This step allows us to work with a straightforward equation without the logarithm.

Step 3: Solve for x

Now solve for \( x \) in the equation: \( 3^2 = x + 2 \). First, calculate \( 3^2 = 9 \), then solve \( 9 = x + 2 \) by subtracting 2 from both sides to get \( x = 7 \).

Step 4: Write the Points

The x-coordinate we found is 7. Since the given y-coordinate is 2, the point on the graph is \( (7, 2) \).

Key Concepts

exponential form

To understand how to convert a logarithmic expression into an exponential form, it's important to grasp what each side of the equation represents. A logarithm, in the expression \( \log_{b}(a) = c \), tells us the power \( c \) to which the base \( b \) must be raised to obtain \( a \). For our case, the expression \( \log_{3}(x+2) = 2 \) tells us that \( 3 \) must be raised to the power of 2 to equal \( x+2 \).
By converting this into an exponential form:

• The base 3 is used as the base of the exponent.
• The right side of the original equation, 2, becomes the exponent.
• Thus, we write \( 3^2 = x+2 \).

Exponential forms make equations without logarithms, making them easier to solve by straightforward algebraic methods.

graphing functions

Graphing functions helps visualize relationships between variables. Here, we focus on the function \( f(x)=\log _{3}(x+2) \). The graph of a logarithmic function is a curve that passes through specific points determined by the values of \( x \). By identifying key characteristics, we can plot it effectively.
Some important aspects of graphing this function include:

• Intercept: Find where \( f(x) \) crosses the x-axis, which happens when \( f(x) = 0 \). For instance, solve \( \log_{3}(x+2) = 0 \).
• Asymptote: Logarithmic graphs never touch the line \( x = -2 \) in this case, as continuing beyond this point would involve taking a log of a non-positive number.
• Behavior: As \( x \) increases, \( f(x) \) rises but slowly.

By graphing, we can determine both where the function passes through certain \( y \)-coordinates and the overall behavior of the function.

solving equations

Solving equations, especially logarithmic ones, might first seem complicated, but using precise methods simplifies this process. Starting with a logarithmic equation like \( \log_{3}(x+2) = 2 \), we proceed by converting to the exponential form, as in previous sections. This leads us to:

• Perform mathematical operations like exponentiation, making handling the equation easier.
• Continuing with solving \( 9 = x + 2 \) reveals the solution is simply reached by basic algebra, subtracting 2 to result in \( x = 7 \).

This step-by-step approach emphasizes systematically isolating \( x \) using valid algebraic steps. Overall, solving logarithmic equations becomes manageable when breaking down the equation by converting forms and applying consistent techniques.