Problem 42 - Precalculus with Limits

Question

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. \(g(x)=\log _{6} x\)

Step-by-Step Solution

Answer

The domain of \(g(x)=\log _{6} x\) is (\(0\), infinity), the x-intercept is at \(x = 1\), and the vertical asymptote is at \(x = 0\). The graph of the function increases slowly to the right, with no upper bound, and decreases without bound as x approaches 0.

Step 1: Finding the Domain

The domain of a logarithm function is always (\(0\),infinity) in the x-axis, because logarithm is only defined for positive numbers. Hence, the domain of \(g(x)=\log _{6} x\) is (\(0\),infinity).

Step 2: Finding the x-intercept

The x-intercept is the value of x when \(g(x) = 0\). We solve the equation \(0 = \log _{6} x\) for x. Use the property of logarithm that states \(a^0=1\), hence \[x = 6^0 = 1\] So, the x-intercept is \(x = 1\).

Step 3: Finding the vertical asymptote

The vertical asymptote of a log function is \(x=0\), because as we approach \(x=0\), the log function value decreases without bound.

Step 4: Sketching the graph

Plot the key points found above, the domain, the x-intercept at (1,0) and vertical asymptote at \(x=0\). The function increases slowly to the right, with no upper bound, and decreases without bound as x approaches 0. The graph of \(g(x)=\log _{6} x\) has these characteristics.

Key Concepts

Domain of Logarithmic Functions

The domain of a logarithmic function defines the set of x-values for which the function is defined. In the case of the function \( g(x) = \log_{6} x \), the domain is limited to positive x-values. This is because the logarithm of zero or a negative number is not defined in the real number system.

For our logarithmic function, the domain is \((0, \infty)\). This interval notation means that x can be any number greater than zero. Here are some key points about the domain of logarithmic functions:

• The base of the logarithm does not affect the domain.
• Logarithmic functions are only defined for positive arguments.
• The domain is always expressed as \((0, \infty)\).

Understanding the domain is essential because it tells us the range of input values for which the function will produce results. Any analysis, such as finding intercepts or plotting the graph, starts by considering the domain.

Understanding X-Intercept of Logarithmic Graphs

The x-intercept of a graph is the point where the graph crosses the x-axis. This means that the y-value or the function value is zero at that point. For the function \( g(x) = \log_{6} x \), finding the x-intercept involves setting the function equal to zero and solving for x.

By the properties of logarithms, we know that if \( \log_{b} x = 0 \), then \( x = b^0 = 1 \). Therefore, for \( \log_{6} x \), the x-intercept is where \( x = 6^0 = 1 \). In other words, the point where the function intersects the x-axis is \((1, 0)\). Let's summarize:

• The x-intercept is found by setting the function equal to zero.
• For logarithmic functions, this usually occurs at \( x = 1 \), regardless of the base.

The x-intercept provides us with a key reference point on the graph and helps us understand the shape and position of the curve relative to the x-axis.

Vertical Asymptote of Logarithmic Functions

A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. For the function \( g(x) = \log_{6} x \), there is a vertical asymptote at \( x = 0 \). This occurs because as x approaches zero from the positive side, the logarithmic value decreases without bound, heading towards negative infinity.

Here’s what you need to remember about vertical asymptotes in logarithmic functions:

• The vertical asymptote is always where \( x = 0 \) for base \( > 1 \).
• This represents the boundary of the domain of the function.
• The curve will decrease indefinitely as it approaches this line but will never intersect it.

Understanding the vertical asymptote helps in drawing the graph since it indicates where the graph will dramatically drop off. This is crucial for correctly sketching the behavior of the logarithmic function. It visually emphasizes that the function is not defined for zero or negative values.