Pure Mathematics | Exponentials and logarithms

Welcome to Exponentials and Logarithms!

Hello future mathematician! This chapter is incredibly important. Exponentials describe growth (like population or compound interest) and logarithms are simply the tools we use to undo that growth and solve those equations. Think of logarithms as the secret key to unlocking the power stored in an exponential equation.

Don't worry if the concepts seem abstract at first; we will break them down step-by-step using clear analogies. Let's conquer P2 together!

Quick Review: The Foundation of Index Laws

Before diving into logs, remember the basic rules of indices (powers). These rules are fundamental, as logarithms are essentially special indices.

• Multiplication Rule: \(a^m \times a^n = a^{m+n}\)
• Division Rule: \(a^m \div a^n = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{mn}\)

1. The Exponential Function and Its Graph

An exponential function has the variable (usually \(x\)) in the power (index). The general form is:

\[y = a^x \quad \text{where } a > 0 \text{ and } a \neq 1\]

Characteristics of the Graph \(y = a^x\)

When \(a > 1\) (e.g., \(y=2^x\)), we see exponential growth:

• The graph passes through the point (0, 1) (since \(a^0 = 1\)).
• The curve is always above the x-axis (since \(a^x > 0\)).
• The x-axis (\(y=0\)) is a horizontal asymptote (the graph gets infinitely close to it but never touches it).
• The graph increases very quickly as \(x\) increases.

Analogy: Imagine a rumor spreading. Initially, it spreads slowly, but then the number of people who know it grows exponentially faster!

2. Introducing the Logarithm: Unlocking the Power

We use logarithms to answer a very specific question: "What power must I raise this base to, to get this number?"

The logarithm is the inverse function of the exponential function.

The Fundamental Definition of a Logarithm

The statement:

\[a^x = N\]

is mathematically equivalent to the logarithmic statement:

\[x = \log_a N\]

(Read as: "x equals log base a of N")

Important Note: The base \(a\) in the exponential form remains the base \(a\) in the logarithmic form.

Memory Aid: The Log Loop

To convert \(x = \log_a N\) back to exponential form, just remember the base \(a\) "loops" around to push \(x\) up into the exponent: \(a\) to the power of \(x\) equals \(N\).

Example Conversion:

• Exponential form: \(2^5 = 32\)
• Logarithmic form: \(\log_2 32 = 5\) (The power is 5)


• Logarithmic form: \(\log_{10} 1000 = 3\)
• Exponential form: \(10^3 = 1000\)

Special Logarithm Values

These rules follow directly from the index laws:

1. Log of the Base: \(\log_a a = 1\) (Because \(a^1 = a\))
2. Log of One: \(\log_a 1 = 0\) (Because \(a^0 = 1\))

3. The Three Laws of Logarithms (The Essential Tools)

These three laws allow us to manipulate and simplify logarithmic expressions. They mirror the index laws.

Law 1: The Product Law (Multiplication becomes Addition)

When you multiply two numbers, their logarithms are added:

\[\log_a (XY) = \log_a X + \log_a Y\]

Think: Since indices multiply by adding powers, logs (which are powers) must add when their results multiply.

Law 2: The Quotient Law (Division becomes Subtraction)

When you divide two numbers, their logarithms are subtracted:

\[\log_a \left(\frac{X}{Y}\right) = \log_a X - \log_a Y\]

Law 3: The Power Law (Powers come down!)

This is arguably the most important law for solving exponential equations. The power on the argument can be moved to the front as a multiplier:

\[\log_a (X^k) = k \log_a X\]

Memory Aid: The Power Law is like the log 'unlocking' the power and bringing it down to ground level where we can work with it!

The Change of Base Formula

Your calculator typically only has buttons for base 10 (\(\log\)) and base \(e\) (\(\ln\)). If you encounter a problem with base 5 (e.g., \(\log_5 10\)), you must use the change of base formula:

\[\log_a N = \frac{\log_b N}{\log_b a}\]

In practice, you always choose \(b=10\) or \(b=e\):

\[\log_5 10 = \frac{\log 10}{\log 5} \quad \text{or} \quad \log_5 10 = \frac{\ln 10}{\ln 5}\]

4. Solving Equations Using Logarithms

When we cannot make the bases the same (e.g., solving \(3^x = 10\)), we must use logarithms.

Step-by-Step Process for Solving \(a^x = b\)

Example: Find \(x\) to 3 significant figures for \(5^x = 40\).

Step 1: Take logarithms of both sides.
Choose any base your calculator handles (base 10 or \(e\)).

\[\log (5^x) = \log (40)\]

Step 2: Apply the Power Law.
Bring the variable \(x\) down to the front.

\[x \log 5 = \log 40\]

Step 3: Isolate \(x\).

\[x = \frac{\log 40}{\log 5}\]

Step 4: Calculate the final value.

\[x \approx \frac{1.602}{0.699} \approx 2.29\]

Solving Equations in Quadratic Form

Some equations look complicated but can be treated like quadratics.

Example: \(3^{2x} - 4(3^x) + 3 = 0\)

Technique: Substitute \(y = 3^x\).

\[(3^x)^2 - 4(3^x) + 3 = 0\] \[y^2 - 4y + 3 = 0\] \[(y - 3)(y - 1) = 0\]

So, \(y=3\) or \(y=1\).

Substitute back:

• Case 1: \(3^x = 3 \implies x=1\)
• Case 2: \(3^x = 1 \implies x=0\)

5. The Exponential Constant \(e\) and Natural Logarithms

There is one special base that pops up everywhere in science, finance, and especially calculus (differentiation and integration). This is the number \(e\).

The definition of \(e\): \(e\) is an irrational number, approximately equal to 2.71828... It is the natural base for continuous growth.

Did You Know? If you compounded interest continuously on \$1, you would end up with \$e after one year.

Natural Logarithms (ln)

When the base of the logarithm is \(e\), we call it the Natural Logarithm, and we use the special notation \(\ln\) (often pronounced "lon").

\[\log_e x \equiv \ln x\]

The laws of logarithms apply perfectly to natural logs as well.

• Key Identity 1: \(\ln e = 1\) (since \(\log_e e = 1\))
• Key Identity 2: \(\ln 1 = 0\)

Solving Equations with \(e\)

To solve an equation involving \(e\), we simply take the natural logarithm (\(\ln\)) of both sides, because \(\ln\) simplifies \(e\).

Example: Solve \(e^{2x-1} = 5\)

1. Take \(\ln\) of both sides: \(\ln(e^{2x-1}) = \ln 5\)
2. Use the Power Law (and the fact that \(\ln e = 1\)): \( (2x-1) \ln e = \ln 5 \implies 2x - 1 = \ln 5\)
3. Isolate \(x\): \(2x = 1 + \ln 5\)
4. Final Answer: \(x = \frac{1 + \ln 5}{2}\)

6. Exponential Modelling: Linearisation

In many real-world applications (like physics or biology), we gather data that seems to follow an exponential relationship. A classic P2 skill is converting these exponential relationships into a straight line so we can easily find the unknown constants using graphs. This process is called linearisation.

Case A: Modelling \(y = A b^x\)

This model relates \(y\) and \(x\) exponentially, where \(A\) and \(b\) are unknown constants. To find \(A\) and \(b\) from experimental data, we must transform the equation into the straight line form \(Y = mX + c\).

Step 1: Take logs of both sides (using any base, but we’ll use \(\log_{10}\)).

\[\log y = \log (A b^x)\]

Step 2: Apply the Product and Power Laws.

\[\log y = \log A + \log (b^x)\] \[\log y = \log A + x \log b\]

Step 3: Map to the straight line form \(Y = mX + c\).

• Y-axis variable (Y): \(\log y\)
• X-axis variable (X): \(x\)
• Gradient (m): \(\log b\) (The gradient of the straight line gives us the constant \(b\))
• Y-intercept (c): \(\log A\) (The intercept gives us the constant \(A\))

If you plot \(\log y\) against \(x\), the data points should fall on a straight line.

Case B: Modelling \(y = A x^n\) (Power Law Relationship)

This model is often confused with Case A, but note that the variable \(x\) is now the base, and \(n\) is a constant power.

Step 1: Take logs of both sides.

\[\log y = \log (A x^n)\]

Step 2: Apply the Product and Power Laws.

\[\log y = \log A + \log (x^n)\] \[\log y = \log A + n \log x\]

Step 3: Map to the straight line form \(Y = mX + c\).

• Y-axis variable (Y): \(\log y\)
• X-axis variable (X): \(\log x\)
• Gradient (m): \(n\) (The gradient directly gives us the power \(n\))
• Y-intercept (c): \(\log A\) (The intercept gives us the constant \(A\))

If you plot \(\log y\) against \(\log x\), the data points should fall on a straight line.

Congratulations! Mastering exponentials and logarithms is crucial for success in P2 and beyond. Keep practicing those law conversions!