Problem 42 - Intermediate Algebra

Question

For Problems \(41-46\), follow the suggested procedure to graph logarithmic functions. (a) Complete the following table and then graph \(f(x)=\) \(\ln x\). (Express the values for \(\ln x\) to the nearest tenth.) \begin{tabular}{c|ccccccc} \(\boldsymbol{x}\) & \(0.1\) & \(0.5\) & 1 & 2 & 4 & 8 & 10 \\ \hline \(\ln x\) & & & & & & & \end{tabular} (b) Complete the following table and express values for \(e^{x}\) to the nearest tenth. \begin{tabular}{c|ccccccc} \(\boldsymbol{x}\) & \(-2.3\) & \(-0.7\) & 0 & \(0.7\) & \(1.4\) & \(2.1\) & \(2.3\) \\ \hline \(\boldsymbol{e}^{\boldsymbol{x}}\) & & & & & & & \end{tabular} Then graph \(f(x)=e^{x}\) and reflect it across the line \(y=x\) to produce the graph for \(f(x)=\ln x\).

Step-by-Step Solution

Answer

Calculate \( \ln x \) and \( e^x \) values, fill the tables, graph them, and use reflection.

Step 1: Calculate \( \ln x \) values

Use a calculator to find the natural logarithm of the given \( x \) values and round them to the nearest tenth:- \( \ln(0.1) \approx -2.3 \)- \( \ln(0.5) \approx -0.7 \)- \( \ln(1) = 0 \)- \( \ln(2) \approx 0.7 \)- \( \ln(4) \approx 1.4 \)- \( \ln(8) \approx 2.1 \)- \( \ln(10) \approx 2.3 \)

Step 2: Fill in the \(\ln x\) Table

Input the calculated values into the table:\[\begin{tabular}{c|ccccccc} \(\boldsymbol{x}\) & 0.1 & 0.5 & 1 & 2 & 4 & 8 & 10 \\hline \ln x & -2.3 & -0.7 & 0 & 0.7 & 1.4 & 2.1 & 2.3 \end{tabular}\]

Step 3: Calculate \( e^x \) values

Use a calculator to compute \( e^x \) values for the given \( x \) values and round them to the nearest tenth:- \( e^{-2.3} \approx 0.1 \)- \( e^{-0.7} \approx 0.5 \)- \( e^{0} = 1 \)- \( e^{0.7} \approx 2 \)- \( e^{1.4} \approx 4.1 \)- \( e^{2.1} \approx 8.2 \)- \( e^{2.3} \approx 10 \)

Step 4: Fill in the \(e^x\) Table

Record these values in the corresponding table:\[\begin{tabular}{c|ccccccc} \(\boldsymbol{x}\) & -2.3 & -0.7 & 0 & 0.7 & 1.4 & 2.1 & 2.3 \\hline \boldsymbol{e}^{\boldsymbol{x}} & 0.1 & 0.5 & 1 & 2 & 4.1 & 8.2 & 10 \end{tabular}\]

Step 5: Graph \(f(x) = \ln x\)

Plot the points from the \( \ln x \) table on a Cartesian plane to graph \( f(x) = \ln x \). Remember that the graph will pass through these points, showing the increasing trend of the natural logarithm function from left to right.

Step 6: Graph \(f(x) = e^x\) and Reflect to Obtain \(f(x) = \ln x\)

First, plot the points obtained for \( e^x \) to graph \( f(x) = e^x \). This graph is the exponential function. Reflect the \( e^x \) graph across the line \( y=x \). This reflection will produce the graph for \( f(x) = \ln x \). The graph of \( \ln x \) will be the inverse of the graph of \( e^x \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is extensively used in mathematics due to its natural properties and appearance in various real-world applications. The natural logarithm of a number answers the question: "To what power must \( e \) be raised in order to obtain this number?"

For example, \( \ln(1) = 0 \) because \( e^0 = 1 \). Similarly, \( \ln(e) = 1 \) because \( e^1 = e \). The logarithmic function \( y = \ln x \) is used to transform multiplication into addition, which simplifies complex calculations.

When graphing \( \ln x \), keep in mind:

• The graph passes through the point (1,0) since \( \ln(1) = 0 \).
• It is undefined for \( x \leq 0 \), so the graph is only in the positive \( x \)-direction.
• The function is increasing, meaning it rises as \( x \) increases.

Understanding the properties and behaviors of the natural logarithm aids in solving problems related to exponential growth and decay, among others.

Exponential Function

An exponential function is characterized by its constant base raised to a variable exponent. The standard form of an exponential function is \( f(x) = a^{x} \), where \( a \) is a positive constant. A particularly significant exponential function is \( f(x) = e^x \), which uses the base \( e \), because of its natural occurrence in mathematics and sciences.

The exponential function \( e^x \) is pivotal in understanding decay and growth processes such as population dynamics, radioactive decay, and interest compounding in finance. One outstanding feature of \( e^x \) is its unique property where the rate of growth of the function is proportional to its current value. This means the more there is, the faster it grows.

When graphing \( e^x \), note:

• The graph passes through the point (0,1) since \( e^0 = 1 \).
• The curve is an increasing function that rises steeply as \( x \) increases.
• This function never touches the \( x \)-axis, as it approaches zero asymptotically on the left.

These characteristics make the exponential function crucial for modeling continuous growth phenomena.

Inverse Functions

Inverse functions "undo" each other. If function \( f \) transforms \( x \) to \( y \), then its inverse \( f^{-1} \) transforms \( y \) back to \( x \). In other words, \( f(f^{-1}(x)) = x \). If \( y = f(x) \) describes a function, \( x = f^{-1}(y) \) describes its inverse.

In the context of exponential and logarithmic functions, \( f(x) = e^x \) and \( f^{-1}(x) = \ln x \) are inverse functions. This means:

• Whatever value \( e^x \) produces, \( \ln \) can take it back to the original number.
• The graph of \( \ln x \) is a reflection of the graph of \( e^x \) over the line \( y=x \).

The geometric relationship between inverse functions is often illustrated by this reflection over the line \( y=x \). This is a helpful concept for visualizing how these functions relate. Inverse functions are essential in solving equations where the variable is in the exponent, enabling us to express the solution in terms of an exponent or a logarithm.