Problem 89 - Precalculus

Question

Use a graphing calculator to plot \(y=\ln x\) and \(y=\frac{\log x}{\log e}\) Are they the same graph?

Step-by-Step Solution

Answer

Yes, they are the same graph.

Step 1: Understand the Functions

We are given two functions: \(y = \ln x\), where \(\ln\) is the natural logarithm (logarithm to base \(e\)), and \(y = \frac{\log x}{\log e}\), where \(\log\) is the logarithm to base 10. Our task is to determine if these functions are identical.

Step 2: Convert the Second Function

Recall that the change of base formula for logarithms states: \(\log_b a = \frac{\log_k a}{\log_k b}\). Applying this to our second function, we rewrite it as follows: \(y = \frac{\log x}{\log e} = \log_e x = \ln x\). This shows that both expressions are indeed equivalent.

Step 3: Plot the Functions with a Graphing Calculator

Input both functions into a graphing calculator or graphing software. Compute \(y = \ln x\) and \(y = \frac{\log x}{\log e}\). You should observe that both produce the same graphical output, confirming they are the same function.

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln x \), is a logarithmic function where the base is the mathematical constant \( e \). The constant \( e \) is approximately equal to 2.71828 and is known as Euler's number. The natural logarithm is commonly used in mathematics due to its natural occurrence in growth processes, such as population growth and compound interest.
The natural logarithm has several important properties:

• \( \ln 1 = 0 \) because \( e^0 = 1 \)
• \( \ln e = 1 \) because \( e^1 = e \)
• \( \ln(xy) = \ln x + \ln y \)
• \( \ln\left(\frac{x}{y}\right) = \ln x - \ln y \)

These properties make it useful in simplifying and solving logarithmic equations. Additionally, the graph of \( \ln x \) increases slowly, smoothly curving upwards, and is only defined for \( x > 0 \).

Logarithm Base 10

Logarithm base 10, often expressed as \( \log x \), operates with a base of 10. This is the logarithmic scale most people are familiar with thanks to its prevalent use in sciences and various fields including pH measurement in chemistry and decibels in acoustics.
Key properties of the base 10 logarithm include:

• \( \log 1 = 0 \) because \( 10^0 = 1 \)
• \( \log 10 = 1 \) because \( 10^1 = 10 \)
• \( \log(10^x) = x \)
• \( \log(xy) = \log x + \log y \)
• \( \log\left(\frac{x}{y}\right) = \log x - \log y \)

The logarithm base 10 graph increases steadily but more gradually compared to the natural logarithm, emphasizing its use in representing large ranges of values in a compact form.

Change of Base Formula

The change of base formula is a crucial tool in logarithmic calculations when you need to convert from one base to another. It can be expressed as \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( b \) is the original base, and \( k \) is the new base we are converting to.
This formula is especially helpful when dealing with calculators which typically only have functions for logarithms base 10 and base \( e \) (the natural log). In the original exercise, we used the change of base formula to convert \( \log x \) to the natural logarithm by rewriting \( y = \frac{\log x}{\log e} \), which simplifies to \( \ln x \), as \( \log_e x \) is the definition of the natural logarithm. This shows how both functions in the exercise were fundamentally the same.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions and verifying mathematical equivalencies, just like in the exercise. By inputting the functions \( y = \ln x \) and \( y = \frac{\log x}{\log e} \) into the calculator, we can graph both functions to see if they coincide.
These devices help to:

• Plot complex functions and observe characteristics such as intersections, slopes, and asymptotes.
• Confirm theoretical solutions by visual approximation.
• Simplify calculations with built-in commands for common mathematical functions.

For students learning about logarithms, graphing calculators provide an immediate, visual way to see how different logarithmic principles and formulas come together to form the same graph, reinforcing understanding.