What is a Graph of y = log x?

S3-SA5-0287

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The graph of y = log x is a special curve that shows how the logarithm of a number 'x' changes as 'x' increases. It helps us visualise the relationship where the output 'y' grows much slower as the input 'x' gets larger.

Simple Example

Quick Example

Imagine you are tracking how many times you have to fold a piece of paper to reach a certain thickness. The number of folds (y) grows very slowly compared to the actual thickness (x) which increases quickly. The graph of y = log x shows a similar pattern: it starts steep but then flattens out, meaning big changes in x lead to small changes in y.

Worked Example

Step-by-Step

Let's plot a few points for y = log x (we'll use log base 10 for simplicity, often written as log x).

Step 1: Understand that log x is only defined for x > 0. So, we can't use x = 0 or negative x values.

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Step 2: Pick some positive values for x and calculate y.
If x = 1, y = log(1) = 0. So, (1, 0) is a point.

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Step 3: Pick another value for x.
If x = 10, y = log(10) = 1. So, (10, 1) is a point.

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Step 4: Pick a value for x between 0 and 1.
If x = 0.1, y = log(0.1) = -1. So, (0.1, -1) is a point.

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Step 5: Pick a larger value for x.
If x = 100, y = log(100) = 2. So, (100, 2) is a point.

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Step 6: Plot these points (0.1, -1), (1, 0), (10, 1), (100, 2) on a graph paper. Connect them with a smooth curve. You will see the curve starts low and increases, but gets flatter as x increases.

Answer: The plotted points (0.1, -1), (1, 0), (10, 1), (100, 2) when connected form the characteristic curve of y = log x.

Why It Matters

Understanding log graphs is crucial in fields like Data Science and AI/ML, where they help analyse data that spans a huge range, like predicting stock market trends or understanding sound intensity. Engineers use them to design systems, and even in Physics, they describe how light and sound behave.

Common Mistakes

MISTAKE: Assuming the graph of y = log x passes through the origin (0,0). | CORRECTION: The logarithm of 0 is undefined. The graph of y = log x never touches or crosses the y-axis, it only gets very close to it as x approaches 0.

MISTAKE: Thinking the graph goes down for very large x values. | CORRECTION: The graph of y = log x always increases as x increases, though it increases very slowly. It never goes down.

MISTAKE: Using negative x values to plot points. | CORRECTION: The logarithm is only defined for positive numbers (x > 0). You cannot find the log of a negative number.

Practice Questions

Try It Yourself

QUESTION: What is the y-intercept of the graph y = log x? | ANSWER: There is no y-intercept, as x must be greater than 0.

QUESTION: For what value of x does the graph of y = log x cross the x-axis? | ANSWER: The graph crosses the x-axis when y = 0. Since log(1) = 0, it crosses at x = 1.

QUESTION: If the graph of y = log x passes through (1000, 3), what does this tell us about the base of the logarithm? | ANSWER: If y = log_b(x), then 3 = log_b(1000). This means b^3 = 1000. So, b = 10. The base of the logarithm is 10.

MCQ

Quick Quiz

Which of these statements is true about the graph of y = log x?

It passes through the origin (0,0).

It is defined for all real numbers x.

It always increases as x increases.

It crosses the y-axis at y = 1.

The Correct Answer Is:

The graph of y = log x is only defined for x > 0, so options A and B are incorrect. It crosses the x-axis at x=1, not the y-axis at y=1, making D incorrect. The graph always increases, though slowly, as x increases.

Real World Connection

In the Real World

In India, when scientists at ISRO track satellite signals, or when sound engineers mix music, they often deal with measurements that vary over a huge range. Logarithmic scales and their graphs help them compress these large ranges into manageable visualisations, making it easier to analyse data like signal strength or sound intensity.

Key Vocabulary

Key Terms

LOGARITHM: A quantity representing the power to which a fixed number (the base) must be raised to produce a given number | BASE: The fixed number in a logarithm (e.g., 10 in log base 10) | X-AXIS: The horizontal line on a graph | Y-AXIS: The vertical line on a graph | ASYMPTOTE: A line that a curve approaches but never touches as it goes to infinity

What's Next

What to Learn Next

Great job understanding the basic graph of y = log x! Next, you can explore 'Transformations of Logarithmic Graphs'. This will help you understand how changing the equation (like y = log (x-2) or y = 2 log x) shifts, stretches, or compresses this fundamental curve.