Problem 129 - Algebra and Trigonometry

Question

In Exercises \(128-131\), graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\ln x, g(x)=\ln x+3 $$

Step-by-Step Solution

Answer

The function \(g(x) = ln(x)+3\) is a vertical transformation of the function \(f(x) = ln(x)\) by 3 units upwards. The shape of the graph remains consistent between \(f\) and \(g\), with only the location of the graph undergoing change.

Step 1: Plotting the Original Function

Start by graphing the original function \(f(x) = ln(x)\). This function cannot be evaluated for \(x \leq 0\), and for \(x > 0\), it represents the natural logarithm of \(x\). For any base of the logarithm, the graph will always cross the x-axis at 1, since the log of 1 in any base always equals zero.

Step 2: Plotting the Transformed Function

Next, plot the graph of the function \(g(x) = ln(x) + 3\). The effect of adding 3 to \(ln(x)\) is a vertical shift upward of the original graph by 3 units. Therefore, without changing the shape of the graph, \(g(x)\) will be the graph of \(f(x)\) shifted three units upwards.

Step 3: Describing the Relationship

After visualizing both graphs, the relationship can be described. The graph of \(g(x)\) is exactly the same as the graph of \(f(x)\) but shifted 3 units upwards. This is a transformation of the original function \(f(x) = ln(x)\) due to the addition of the constant 3.

Key Concepts

Logarithmic Transformations

Logarithmic transformations involve manipulating the equation of a logarithmic function to create a new function with different characteristics. This process often involves altering certain parameters of the logarithmic equation. In our exercise, we consider two functions: \(f(x) = \ln(x)\) and \(g(x) = \ln(x) + 3\).

The core transformation here is the addition of a constant to the natural logarithm function to form a new function. By adding a constant, we effectively perform a vertical shift (which we'll examine in the next section). But it's important to understand that the base graph remains unchanged in terms of its growth trend and shape: it still has the same basic curve of the natural logarithm.

When doing logarithmic transformations, make sure to note:

• The location of key points (like the x-intercept) might change.
• The overall trend of the curve maintains its logarithmic nature.
• Transformations can give clues about quicker ways to graph functions without recalculating the whole table of values.

Understanding transformations like these helps in analyzing how different algebraic manipulations affect a function's graph.

Vertical Shifts

A vertical shift moves the entire graph of a function up or down in relation to the y-axis by adding or subtracting a constant value. For logarithmic functions, this shift is clearly apparent in the problem given, where \(g(x) = \ln(x) + 3\) is obtained from \(f(x) = \ln(x)\).

In terms of graphing, this means:

• The y-values increase by 3 units if you were to look straight up from the starting f(x) value.
• The shape of the graph remains the same.
• If \(f(x)\) crosses the x-axis at \(x = 1\), \(g(x)\) will also intercept the line y = 3 units up from the x-axis.

Vertical shifts are useful for expressing transformations in real-world contexts, such as translating physical movements or adjusting scales. When graphing new functions or solving equations, recognizing a vertical shift can save time and effort by allowing you to sketch graphs more quickly and with fewer calculations.

Natural Logarithm Properties

The natural logarithm, symbolized by \(\ln\), is a logarithmic function with a specific base \(e\), where \(e\approx 2.718\). It has many useful mathematical properties that make it preferable in advanced calculus and applied settings.

Some important properties are:

• \(\ln(1) = 0\). This property is why the graph intersects the x-axis at 1.
• For any positive real number \(x\), \(\ln(x)\) is defined. Thus, the graph exists only for \(x > 0\).
• \(\ln(ab) = \ln(a) + \ln(b)\) showcases the additive property useful in algebraic simplifications.
• \(\ln(a^b) = b\ln(a)\) is helpful in solving exponential equations.
• The derivative of \(\ln(x)\) is \(1/x\), which is central to calculus principles.

Understanding these helps to leverage the natural logarithm's properties in various equations and transformations, streamlining complex mathematical manipulations by using their well-organized characteristics.