Math Problem Statement

Graph the logarithmic function \( f(x) = 2 + \log_3{x} \). Plot two points on the graph of the function, and also draw the asymptote.

Solution

The function given is:

f(x)=2+log3xf(x) = 2 + \log_3{x}

This is a logarithmic function with a vertical shift. Let's analyze it step-by-step.

1. Understanding the Function

The base of the logarithm is 3, so the function log3x\log_3{x} has a basic shape of a logarithmic curve, which generally starts from x=0x = 0 (approaching a vertical asymptote there) and increases slowly as xx grows. The "+ 2" means the entire graph of log3x\log_3{x} is shifted up by 2 units.

2. Vertical Asymptote

For the function f(x)=2+log3xf(x) = 2 + \log_3{x}, the vertical asymptote is at x=0x = 0, since logarithmic functions are undefined for x0x \leq 0. This asymptote will be a vertical line at x=0x = 0.

3. Plotting Points

Let's find two points on the graph by substituting values for xx:

  • For x=1x = 1: f(1)=2+log31=2+0=2f(1) = 2 + \log_3{1} = 2 + 0 = 2 So, one point is (1,2)(1, 2).

  • For x=3x = 3: f(3)=2+log33=2+1=3f(3) = 2 + \log_3{3} = 2 + 1 = 3 So, another point is (3,3)(3, 3).

Summary of Key Information

  • Vertical Asymptote: x=0x = 0
  • Points on the Graph: (1,2)(1, 2) and (3,3)(3, 3)

Would you like a graph of this function or more details on any part of the solution?


Related Questions

  1. What is the effect of changing the base of the logarithm on the graph?
  2. How does the function f(x)=2+logbxf(x) = 2 + \log_b{x} change if we add a coefficient to logbx\log_b{x}?
  3. What is the domain and range of f(x)=2+log3xf(x) = 2 + \log_3{x}?
  4. How would we find the x-intercept of f(x)=2+log3xf(x) = 2 + \log_3{x}?
  5. What happens to the graph if we replace log3x\log_3{x} with log3(xh)\log_3{(x - h)} for some constant hh?

Tip

For logarithmic functions of the form f(x)=logbx+cf(x) = \log_b{x} + c, the constant cc shifts the graph vertically, and the base bb affects the rate of growth.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Transformations of Functions
Vertical Shifts
Vertical Asymptotes

Formulas

f(x) = c + \log_b{x}

Theorems

Properties of Logarithmic Functions

Suitable Grade Level

Grades 10-12