The Logarithmic Function (Extended Content E3.7)

Hello mathematicians! This chapter introduces you to the fascinating world of logarithms. Don't worry if the name sounds complicated—logs are simply the mathematical way of answering a question about powers (or indices). Since the logarithmic function is the inverse of the exponential function, mastering this topic helps you solve problems related to fast growth, like compound interest or population change. Let's conquer it together!

1. Logarithms and Exponentials: The Inverse Relationship

The core concept of logarithms is that they undo exponential functions.

What is an Exponential Function?

An exponential function looks like: \(y = a^x\).
Here, \(a\) is the base, and \(x\) is the exponent (or power).
Example: \(10^2 = 100\). (The base 10 raised to the power 2 equals 100.)

What is a Logarithmic Function?

A logarithm asks: "What power do I need to raise the base to, to get this number?"

The logarithmic function \(x = \log_a y\) is the inverse of the exponential function \(y = a^x\).

Memory Aid: The Logarithmic Switcheroo

You must be able to switch easily between the two forms:

  • Exponential Form: \(y = a^x\)
  • Logarithmic Form: \(x = \log_a y\)


Analogy: Think of the base \(a\) as the foundation. In the exponential form, the foundation holds up the exponent \(x\). When you switch to log form, the base \(a\) still acts as the foundation, but it moves to a subscript position, asking for the exponent \(x\).

Step-by-Step Conversion:

  1. Start with the Base: This is the number that stays small (subscript). (In \(a^x\), it’s \(a\).)
  2. Switch the Power and the Answer: The exponent \(x\) becomes the result of the log, and the answer \(y\) becomes the input (the number you are taking the log of).

Example 1: Convert \(2^3 = 8\) to log form.

The base is 2. The power is 3. The answer is 8.
Log form: \(3 = \log_2 8\)
(Read as: "Log base 2 of 8 equals 3")

Example 2: Convert \(4 = \log_{10} 10000\) to exponential form.

The base is 10. The power is 4. The answer is 10000.
Exponential form: \(10^4 = 10000\)

Quick Review: The Inverse Concept

If we have a function \(f(x) = a^x\), its inverse function is \(f^{-1}(x) = \log_a x\).

The input of the exponential function is the power (\(x\)), and the output is the answer (\(y\)).
The input of the logarithmic function is the answer (\(y\)), and the output is the power (\(x\)).

2. Logarithmic Notation and Base 10

In your IGCSE studies, when working with logarithmic equations, you will generally rely on your calculator, which uses a specific base.

Base 10 Logarithms

The syllabus confirms that all logs will be base 10 unless otherwise stated.

When you see the notation \(\log x\) written on its own (without a small subscript number), it means:

$$\log x = \log_{10} x$$

Your calculator has a button labelled 'LOG'. This button calculates the logarithm base 10.

Example: \(\log 100\) (or \(\log_{10} 100\)) asks: "What power do I raise 10 to, to get 100?"
Answer: 2, because \(10^2 = 100\).

Key Takeaway: Log base 10 is the standard log used for general calculations and is directly accessible on your calculator.

3. Solving Exponential Equations Using Logarithms

Logarithms are essential tools for solving equations where the unknown variable (\(x\)) is in the exponent (power).

Imagine you need to solve: \(a^x = b\). How do you find \(x\)? You use logarithms!

The Change of Base Rule (for calculation)

To solve any exponential equation of the form \(a^x = b\), the value of \(x\) can be found using the following formula:

$$x = \frac{\log b}{\log a}$$

This formula uses base 10 logarithms (or any common base, but we use base 10 because of the calculator).

Step-by-Step Example: Solve \(5^x = 30\)

  1. Identify a and b:
    \(a\) (the base) = 5
    \(b\) (the answer) = 30
  2. Apply the formula:
    $$x = \frac{\log 30}{\log 5}$$
  3. Use your graphic display calculator (GDC):
    \(x \approx \frac{1.477}{0.699}\)
  4. Calculate the final answer:
    \(x \approx 2.11\) (to 3 significant figures)

Check: Does \(5^{2.11}\) roughly equal 30? Since \(5^2 = 25\) and \(5^3 = 125\), 2.11 is a very reasonable answer!
(Remember to use the exact log values in your calculator and only round the final answer, as required for accuracy.)

Common Mistake Alert!

Do not confuse \(\frac{\log b}{\log a}\) with \(\log(\frac{b}{a})\). They are very different!

$$ \frac{\log b}{\log a} \neq \log b - \log a $$

Always calculate the log of the numerator and the log of the denominator separately, then divide the results.

Key Takeaway: Use the change of base formula to find unknown powers in exponential equations.

4. Applications of Logarithms in Real-World Problems

Logarithms allow us to find the time it takes for growth or decay processes to reach a specific level. These problems often appear in the context of compound interest or exponential growth and decay.

Example: Compound Interest (Finding Time)

Imagine you invest $1000 at 5% compound interest per year. You want to know how many years (\(t\)) it takes for your investment to reach $1500.
The formula for compound interest is usually: $$A = P(1 + r)^t$$

  • \(A\) (Final amount) = 1500
  • \(P\) (Principal) = 1000
  • \(r\) (Rate, as a decimal) = 0.05
  • \(t\) (Time in years) is the unknown.

Step 1: Set up the exponential equation.
$$1500 = 1000(1 + 0.05)^t$$ $$1500 = 1000(1.05)^t$$

Step 2: Isolate the exponential term.
$$\frac{1500}{1000} = (1.05)^t$$ $$1.5 = 1.05^t$$

Step 3: Solve for the exponent (t) using logs.
We use the structure \(a^x = b \implies x = \frac{\log b}{\log a}\), where \(a = 1.05\), \(b = 1.5\), and \(x = t\).

$$t = \frac{\log 1.5}{\log 1.05}$$

Step 4: Calculate using the GDC.
$$t \approx \frac{0.17609}{0.021189}$$ $$t \approx 8.31 \text{ years}$$

Did You Know?

Logarithms are used extensively in fields like geology (measuring earthquake magnitude on the Richter scale) and acoustics (measuring sound intensity in decibels, which is a logarithmic scale)! They are perfect for handling numbers that span a huge range.

5. Graphing Exponential and Logarithmic Functions

While the syllabus doesn't require complex sketching of the log function by hand, understanding the basic shape and its relationship to the exponential function is vital, especially when using your GDC (Graphic Display Calculator).

Graph of the Exponential Function (\(y = a^x\), where \(a>1\))

  • It passes through the point \((0, 1)\) (since \(a^0 = 1\)).
  • It increases rapidly (exponential growth).
  • The x-axis (\(y=0\)) is a horizontal asymptote (the graph gets closer but never touches).

Graph of the Logarithmic Function (\(y = \log_a x\), where \(a>1\))

Since the log function is the inverse, its graph is a reflection of \(y = a^x\) across the line \(y = x\).

  • It passes through the point \((1, 0)\) (the inverse of \((0, 1)\)).
  • It increases slowly (especially for large \(x\)).
  • The y-axis (\(x=0\)) is a vertical asymptote.
  • Important: The domain (x-values) is \(x > 0\). You cannot take the log of zero or a negative number!

Using your GDC (as required in E3.2), you should be able to sketch the log graph, produce a table of values, and find the intersection points with other graphs to solve equations.

6. Summary and Key Takeaways

The Logarithm Checklist

  • Definition: A logarithm is an exponent. \(x = \log_a y\) means \(a^x = y\).
  • Inverse: The logarithmic function is the inverse of the exponential function.
  • Base 10: If no base is written, assume base 10: \(\log x = \log_{10} x\).
  • Solving Equations: To solve \(a^x = b\), always use the formula: $$x = \frac{\log b}{\log a}$$
  • Asymptote: The graph of \(y = \log x\) has an asymptote at \(x=0\) (the y-axis).

You have successfully tackled the definition and key application of logarithms! Practice converting between the two forms, and you will find solving exponential problems much easier. Keep up the great work!