Exponents and logarithms

Introduction to Exponents and Logarithms

Welcome to one of the most powerful tools in your mathematical toolkit! In this chapter, we explore how to handle numbers that grow or shrink very quickly. Whether it’s the spread of a virus, the growth of a bank account, or the way sound levels are measured, exponents and logarithms are the language of change. Don’t worry if these look intimidating at first—once you learn the "rules of the game," they become much easier to manage.

1. Laws of Exponents (SL & AHL)

An exponent (or power) tells us how many times to multiply a number by itself. In the expression \(a^n\), \(a\) is the base and \(n\) is the exponent.

The Fundamental Rules

For the SL course, you need to be comfortable with integer exponents (whole numbers and their negatives). Here are the rules you must know:

1. Multiplication Rule: When multiplying the same base, add the exponents.
\(a^m \times a^n = a^{m+n}\)
Example: \(2^3 \times 2^4 = 2^7\)

2. Division Rule: When dividing the same base, subtract the exponents.
\(\frac{a^m}{a^n} = a^{m-n}\)
Example: \(\frac{5^6}{5^2} = 5^4\)

3. Power of a Power: When raising a power to another power, multiply the exponents.
\((a^m)^n = a^{m \times n}\)
Example: \((3^2)^4 = 3^8\)

4. The Zero Power: Any non-zero number raised to the power of zero is 1.
\(a^0 = 1\)
Example: \(157^0 = 1\)

5. Negative Exponents: A negative exponent indicates a reciprocal (flip it!).
\(a^{-n} = \frac{1}{a^n}\)
Example: \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)

Rational Exponents (AHL Only)

For AHL students, exponents can also be fractions. This is just another way of writing roots.
\(a^{\frac{1}{n}} = \sqrt[n]{a}\)
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\)

Example: \(16^{\frac{1}{2}} = \sqrt{16} = 4\).
Example: \(8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4\).

Quick Tip: If you see a fraction as an exponent, the bottom number is the root and the top number is the power!

2. Introduction to Logarithms (SL & AHL)

Logarithms are simply the inverse of exponents. If an exponent asks "What do I get when I multiply this?", a logarithm asks "What power did I use to get this number?"

The relationship is: \(b^x = y \iff \log_b(y) = x\)

Example: Since \(10^2 = 100\), we can say \(\log_{10}(100) = 2\).

Two Special Bases

In this course, you will primarily work with two specific bases:

1. Base 10 (Common Logarithm): Written as \(\log_{10}(x)\) or often just \(\log(x)\). This is the standard on most calculators.
2. Base \(e\) (Natural Logarithm): The number \(e\) is a mathematical constant approximately equal to \(2.718\). The logarithm with base \(e\) is written as \(\ln(x)\).

Using Technology: In the Applications and Interpretation course, you are expected to use your Graphic Display Calculator (GDC) to evaluate logarithms. Look for the "log" and "ln" buttons on your device.

Key Takeaway: Logarithms "pull down" exponents, making them very useful for solving equations where the unknown variable is in the power.

3. Laws of Logarithms (AHL Only)

AHL students must be able to manipulate and simplify expressions using the Laws of Logarithms. These rules are very similar to the exponent rules!

1. The Product Rule: \(\log_a(xy) = \log_a(x) + \log_a(y)\)
(Think: Multiplication inside becomes addition outside.)

2. The Quotient Rule: \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\)
(Think: Division inside becomes subtraction outside.)

3. The Power Rule: \(\log_a(x^m) = m \log_a(x)\)
(This is the most famous one! You can move the exponent to the front as a multiplier.)

Common Mistake to Avoid: \(\log(x + y)\) is NOT the same as \(\log(x) + \log(y)\). There is no rule for adding or subtracting inside the logarithm!

4. Scaling and Real-World Contexts

In the real world, we often use logarithms to handle numbers that are either incredibly large or incredibly small. This is often called logarithmic scaling.

Did you know? The Richter scale (for earthquakes), the decibel scale (for sound), and the pH scale (for acidity) are all logarithmic. If an earthquake moves from Magnitude 4 to Magnitude 5, it is actually 10 times more powerful, not just "1 unit" more!

Linearization (AHL Connection)

AHL students will use logarithms to turn curved graphs into straight lines (linearizing data). By taking the log of data points, we can identify exponential relationships more easily. You will explore this further in the Functions (Topic 2) section.

Summary Checklist

Before moving on, make sure you can:

• Use the laws of exponents to simplify expressions (SL/HL).
• Convert between exponential form \(b^x = y\) and logarithmic form \(\log_b(y) = x\) (SL/HL).
• Use your GDC to find the value of \(\log(x)\) and \(\ln(x)\) (SL/HL).
• Apply the product, quotient, and power laws of logs to expand or condense expressions (AHL).
• Simplify expressions that include roots using rational exponents (AHL).

Note: For more on how these are used to solve growth and decay problems, see the chapter on "Financial mathematics" or "Modelling exponential functions."