Problem 47 - Precalculus with Limits

Question

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. \(y=\log \left(\frac{x}{7}\right)\)

Step-by-Step Solution

Answer

The domain of the function is \(x > 0\), and the vertical asymptote is at \(x = 0\). The x-intercept is at \(x = 7\).

Step 1: Find the Domain

The domain of the logarithmic function \(y = \log(x/7)\) is the set of all 'x' values for which \(\frac{x}{7} > 0\). To find the domain, we set \(\frac{x}{7} > 0\), which simplifies to \(x > 0\). Thus, the domain for this function is \(x > 0\).

Step 2: Calculate the x-intercept

To find the x-intercept, we set \(y = 0\) and solve for 'x'. This gives us the equation \(0 = \log(x/7)\). Using the inverse property of logarithms, we rewrite this as \(1 = \frac{x}{7}\), which simplifies to \(x = 7\). So, the x-intercept is 7.

Step 3: Identify the Vertical Asymptote

The vertical asymptote for the function \(y = \log(x/7)\) is the vertical line that the graph approaches as x approaches 0 from the right. This is always \(x = 0\) for logarithmic functions without any horizontal shifts.

Step 4: Graph the Function

To sketch the graph of \(y = \log(x/7)\), plot the x-intercept at \(x = 7\) and draw a curve from this point towards infinity, getting steeper as it gets further away from 0. The curve should approach but never touch the y-axis, which is the vertical asymptote at \(x = 0\).

Key Concepts

Domain of a Function

The domain of a function is a crucial concept in mathematics as it specifies which input values are permissible for the function. In the case of logarithmic functions, like the one given by \[ y = \log\left(\frac{x}{7}\right) \]there are specific rules. For a logarithmic function, the expression inside the log must always be positive. This ensures that we're not trying to take the log of zero or a negative number, which are undefined in the real number system.

In our exercise, to determine the domain, we set the argument of the logarithm greater than zero:\[ \frac{x}{7} > 0 \]This inequality simplifies to \(x > 0\). Thus, the domain of this function is all positive real numbers. Practically, this means that you can input any positive number for \(x\), but any negative number or zero will not work. Understanding this ensures that we work within the realm where the function is actually defined and can provide meaningful outputs.

X-Intercept

The x-intercept of a function's graph is the point where the graph crosses the x-axis. It's a significant point because at this point, the output of the function, \(y\), is zero. To find the x-intercept for the function \[ y = \log\left(\frac{x}{7}\right) \],we set \(y = 0\) and solve for \(x\):\[ 0 = \log\left(\frac{x}{7}\right) \]By applying the properties of logarithms, particularly the inverse function rule, we convert this equation into:\[ 1 = \frac{x}{7} \]Solving for \(x\) yields: \[ x = 7 \]Thus, the x-intercept of the function is at the point \((7, 0)\). This means the graph crosses the x-axis at this point, providing a pivot point where the graph changes direction.

Vertical Asymptote

The concept of a vertical asymptote is often encountered with logarithmic functions. A vertical asymptote is a line that the graph of a function approaches as the input (usually \(x\)) nears a certain value, but never actually reaches. For the function \[ y = \log\left(\frac{x}{7}\right) \],the vertical asymptote is crucial because it indicates where the function grows without bounds as \(x\) approaches this critical value from the positive side.
For logarithmic functions like ours, the vertical asymptote typically occurs where the argument of the logarithm equals zero. Setting the argument equal to zero gives us:\[ \frac{x}{7} = 0 \]Solving this equation leads to the point:\[ x = 0 \]This tells us the vertical asymptote for this function is along the line \(x = 0\). On a graph, you'll notice that the curve gets closer and closer to this line, but never quite hits it, representing the boundary of our function's behavior as the values get exceedingly smaller yet positive.