Problem 84 - Algebra and Trigonometry

Question

The Height of the Graph of a Logarithmic Function Suppose that the graph of \(y=2^{x}\) is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 \(\mathrm{mi}\) (b) If the graph of \(y=\log _{2} x\) is drawn on the same set of axes, how far to the right of the origin do we have to go before the height of the curve reaches 2 \(\mathrm{ft} ?\)

Step-by-Step Solution

Answer

(a) 265 miles; (b) about 265 miles.

Step 1: Convert Units for Distance

First, convert the distance from feet to inches because the graph uses inches as the unit of measurement. Given: 2 feet = 24 inches.

Step 2: Calculate Height of the Graph at x=24

Substitute \(x = 24\) into the function \(y = 2^x\) to calculate the height: \(y = 2^{24}\).

Step 3: Convert Calculation Result to Miles

Calculate \(2^{24}\), which equals 16,777,216 inches. Convert this value to miles knowing that there are 63,360 inches in a mile: \(\frac{16,777,216}{63,360} \approx 265 \) miles.

Step 4: Find x-Value for Logarithmic Curve Height

To find where the height of the graph of \(y = \log_2 x\) is 2 feet (or 24 inches), set \(\log_2 x = 24\). Convert this equation to exponential form: \(x = 2^{24}\).

Step 5: Interpretation of x-value for Logarithmic Graph

Since \(x = 2^{24}\) was previously calculated, it equals 16,777,216. Therefore, you need to go about 16,777,216 inches to the right, which converts to approximately 265 miles.

Key Concepts

Exponential Function

An exponential function, like the one in this problem, is a mathematical function where a constant base is raised to a power or exponent. In this case, we are exploring the function \(y = 2^x\). This indicates that whatever value is selected for \(x\), the function's output is "2" raised to that power.
Exponential functions are fascinating because they model rapid growth or decay.

• In our example, if \(x = 1\), then the function outputs \(y = 2^1 = 2\).
• If \(x = 2\), it becomes \(y = 2^2 = 4\).
• As \(x\) increases, \(y\) grows at an accelerating pace.

This property makes exponential functions incredibly useful in real-world contexts, such as calculating population growth or compound interest in finance.
They visually resemble a J-shaped curve ascending steeply on a graph, especially as \(x\) increases.

Convert Units

Sometimes, it's necessary to convert units to solve a problem correctly and maintain consistency.
In this exercise, we're required to convert units from feet to inches and from inches to miles.
For distance, we know:

• 1 foot = 12 inches
• 1 mile = 63,360 inches

When a problem involves multiple units, ensuring everything is in alignment is key to getting the right answer.
In our case, starting with 2 feet and converting it to inches gives 24 inches.
Similarly, once we have a distance in inches, we can calculate its equivalent in miles, like converting 16,777,216 inches to about 265 miles.

Coordinate Plane

When we plot functions such as \(y = 2^x\) or \(y = \log_2 x\) on a coordinate plane, it helps us visualize their behavior.
A coordinate plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing us to chart the relationship between two variables.

• The x-axis generally represents the input values.
• The y-axis shows the calculated values or outputs.

This setup provides an effective way to understand how these mathematical functions grow and change.
In our example, we observe that as we move right on the x-axis, our exponential graph climbs rapidly vertically, showing expansive growth.
The logarithmic graph, meanwhile, gradually increases, displaying slower growth over a vast distance.

Logarithm Base 2

Logarithms are the inverse operations of exponentiation.
When dealing with a logarithm base 2, we're asking, "To what power must 2 be raised to achieve a certain number?" In mathematical terms, for the logarithm \(\log_2 x = y\), it means that \(2^y = x\).In this exercise's context, we want to know the value of \(x\) such that the graph reaches a height of 24 inches.
This forms the equation \(\log_2 x = 24\), which can be rewritten in exponential format as \(x = 2^{24}\).

• The answer to this is a substantial number, 16,777,216.
• This demonstrates how logarithmic growth continues increasing but at a decreasing rate compared to exponential growth.

Logarithmic functions like \(y = \log_2 x\) are frequently used in fields such as information technology and science to quantify processes occurring over a wide range of scales.