Question

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln (x-2) $$

Short answer

Shift \(y = \ln x\) 2 units right for \(y = \ln(x - 2)\).

Step-by-step solution

Understand the Parent Function

The parent function here is \(y = \ln x\). It is a logarithmic function which is only defined for \(x > 0\) and passes through the point \((1, 0)\). As \(x\) approaches \(0^+\), the function \( \ln x \) approaches \(-\infty\).

Identify the Transformation

The given function is \(y = \ln(x - 2)\). This represents a horizontal shift of the graph of \(y = \ln x\) to the right by 2 units.

Determine the New Domain

Since the function is \(y = \ln(x - 2)\), it's defined for \(x - 2 > 0\), thus, \(x > 2\). This means the domain of the new function is \((2, \infty)\).

Find the New Intercepts

To find the x-intercept, set \(y = 0\):\[ \ln(x - 2) = 0 \]Solving this, you get \(x - 2 = e^0 = 1\), hence \(x = 3\). Thus, the graph passes through \((3, 0)\).

Sketch the Graph

Transpose the basic shape of the \(y = \ln x\) graph to the right by 2 units. The new graph should approach \(-\infty\) as \(x\) approaches 2 from the right, and continue rising as \(x\) increases. The vertical asymptote is now at \(x = 2\) instead of \(x = 0\).

Key concepts

Logarithmic Functions

Logarithmic functions are a type of mathematical function characterized by their rapid growth and the presence of a logarithm in the formula. The most common form is the natural logarithm, represented as \(y = \ln x\). This notation means the power to which the base \(e\) (approximately 2.718) must be raised to produce the number \(x\). One key property of logarithmic functions is that they are only defined for positive values of \(x\).

These functions are important in many fields such as biology, economics, and physics, because they model processes that change rapidly at first and then slowly even out. For example, they can describe the rate of growth of populations or the cooling of substances. Understanding how to manipulate and transform these functions is crucial as it allows us to adapt them for different scenarios. By knowing the shape and behavior of the basic logarithm \(y = \ln x\), we can anticipate changes when the function is transformed.

Horizontal Shifts

Horizontal shifts modify the position of a graph along the x-axis without affecting its vertical orientation. For the function \(y = \ln(x - 2)\), this represents a horizontal shift. In general, subtracting a number inside the function argument, as in \(x - 2\), moves the graph to the right by that number of units.

If the term was added, such as in \(y = \ln(x + 2)\), it would indicate a shift to the left by 2 units instead. The basic shape of the logarithmic function remains intact, but its starting point on the x-axis changes. Such understanding is essential in sketching complex graphs since it allows predicting how various transformations interact.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For logarithmic functions, there is a vertical asymptote that arises due to their undefined nature at certain x-values. For the parent function \(y = \ln x\), the asymptote is at \(x = 0\) since the function is not defined for \(x \leq 0\).

When horizontal shifts occur, such as moving to \(y = \ln(x - 2)\), the vertical asymptote shifts as well. Rather than being at \(x = 0\), it moves to \(x = 2\). Recognizing this shift is key to understanding where the new graph will approach negative infinity, as it gives insight into the behavior of the function near that x-value.

Domains of Functions

The domain of a function refers to the complete set of possible input values (x-values) over which the function is defined. For logarithmic functions, determining the domain is crucial because it guides you on the permissible values of \(x\). The function \(y = \ln x\) is defined for \(x > 0\), meaning its domain is \((0, \infty)\).

When a horizontal shift is applied, such as in \(y = \ln(x - 2)\), the domain is also affected. The condition \(x - 2 > 0\) needs to be satisfied. Solving this inequality gives \(x > 2\), thus shifting the domain to \((2, \infty)\). This modification in domain reflects the horizontal shift in the graph and defines the new starting point for the function.