Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=\log _{10}(x-1)$$

Step-by-Step Solution

Answer

The domain of the function \(y=\log_{10}(x-1)\) is \(x \gt 1\). The vertical asymptote is \(x = 1\), and the x-intercept occurs when \(x = 2\). The graph of the function \(y=\log_{10}(x-1)\) intersects the \(x\)-axis at \(x = 2\), and becomes infinitely close to the line \(x = 1\) as \(x\) approaches 1 from the right, but does not touch or cross it.

Step 1: Find the domain

The domain of a logarithmic function is the set of all real numbers for which the logarithm is defined. In our case, we are looking at the function \(y=\log_{10}(x-1)\). The logarithm is defined for \(x-1 \gt 0\). If we solve this inequality, we get \(x \gt 1\). So the domain of the function is \(x \gt 1\).

Step 2: Determine the vertical asymptote

The vertical asymptote of a logarithmic function occurs where the argument of the logarithm is zero. In this case, we set \(x-1 = 0\) and solve for \(x\). The solution to this equation is \(x = 1\). Therefore, the vertical asymptote of the given logarithm function is \(x = 1\).

Step 3: Identify the x-intercept

The x-intercept of a function is the point where the graph of the function intersects the x-axis. The x-intercept occurs when \(y = 0\). When \(y=0\), this implies that \(\log_{10}(x-1) = 0\). Using the property of logarithms that says \(\log_b A = C\) is the same as \(B^C = A\), we find \(10^0 = (x - 1)\), which simplifies to \(1 = x - 1\). Solving this equation gives \(x = 2\), which is the x-intercept of the function.

Step 4: Sketch the graph

With the values we have determined, we start by drawing a vertical line at \(x = 1\), which is the vertical asymptote. The graph approaches this line but never crosses it. The graph will cross the x-axis at \(x = 2\). The graph will increase as \(x\) gets larger, and the curve will get closer and closer to the vertical asymptote as \(x\) gets closer to 1 from the right. The graph will be in Quadrant I.

Key Concepts

Domain of a Logarithmic Function

Logarithmic functions are fascinating subjects that perplex many students, but understanding their domain is the first step in demystifying them.
The domain of a logarithmic function refers to the set of all possible input values (often denoted as 'x' values) for which the function is defined. With our example (y=log _{10}(x-1))

• The function (log _{10}) only accepts positive arguments, because you can't take the log of zero or a negative number - it's mathematically undefined.
• By examining (x-1> 0), we know that the function is undefined for (x ≤ 1). This makes the domain { x | x > 1 } (in set-builder notation), or simply all real numbers greater than 1.

Vertical Asymptote

Dive into the concept of vertical asymptotes, and you delve into the heart of what gives a logarithmic graph its distinct shape.
The vertical asymptote of a logarithmic function like (y=log _{10}(x-1))

• It occurs where the function heads off towards infinity, which in simpler terms means 'it goes up or down indefinitely' near a certain value of (x).
• In our case, as (x approaches 1 from the right, the (y) value becomes infinitely large. That's because the logarithmic function is undefined when its input is zero – hence the line (x = 1) becomes our vertical asymptote.
• It's a concept that often feels abstract, but it's essentially where the function is infinitely sensitive to changes around a certain point, in this case, (x = 1).

X-intercept

The (x-intercept is where a function crosses the (x-axis, which is the horizontal line that runs left to right on a graph. For logarithmic functions, figuring out where this intersection occurs is as much about understanding logarithms as it is about solving equations.
The (x-intercept for our function (y=log _{10}(x-1)) is found by setting (y to 0 and solving for (x):

• When (y=0), (log_{10}(x-1) = 0), which indicates that (x-1 ) must be 1 (since 10^0 = 1).
• Therefore, adding 1 to both sides of the equation (1 = x - 1), we find that (x = 2); this is where the function will intersect the (x-axis.

It's an important value to calculate because it's one of the critical points that help outline the shape and position of the graph.