Mathematics | Exponentials and logarithms

Welcome to Exponentials and Logarithms!

Hello future mathematician! This chapter, Exponentials and Logarithms, is one of the most fundamental and useful areas of P2 Pure Mathematics. Don't worry if the symbols look strange at first—logarithms are simply the mathematical tool we use to "undo" an exponential function.

Why are they important? Exponentials model almost every form of growth or decay in the real world, from compound interest in finance to radioactive decay and population dynamics. Mastering these tools gives you the power to analyze these changes!

1. Revisiting the Index Laws (The Foundation)

Before diving into exponentials, we need to be crystal clear on the rules of indices (powers). These laws are the bedrock of the entire chapter.

The Key Index Laws

Let \(a\) and \(b\) be positive numbers, and \(m\) and \(n\) be real numbers.

• Law 1: Multiplication (Adding powers)
  \(a^m \times a^n = a^{m+n}\)
• Law 2: Division (Subtracting powers)
  \(\frac{a^m}{a^n} = a^{m-n}\)
• Law 3: Power of a Power (Multiplying powers)
  \((a^m)^n = a^{mn}\)
• Law 4: Zero Index
  \(a^0 = 1\) (Anything raised to the power zero is 1.)
• Law 5: Negative Index (The Reciprocal)
  \(a^{-n} = \frac{1}{a^n}\)
• Law 6: Fractional Index (Roots)
  \(a^{1/n} = \sqrt[n]{a}\)
• Law 7: General Fractional Index
  \(a^{m/n} = (\sqrt[n]{a})^m\)

Quick Review Tip: When solving index equations, always try to make the bases the same first.
Example: To solve \(2^x = 32\), recognize that \(32 = 2^5\). Therefore, \(2^x = 2^5\), so \(x=5\).

2. Exponential Functions \(y = a^x\)

An exponential function is any function where the variable (\(x\)) is in the exponent (power). The number \(a\) is called the base, and we require \(a > 0\).

Key Characteristics of \(y = a^x\)

• The Base: If \(a > 1\), the function represents exponential growth (it gets steeper very quickly).
• The Base: If \(0 < a < 1\), the function represents exponential decay (it decreases quickly but never reaches zero).
• Intercept: All graphs of \(y = a^x\) pass through the point \((0, 1)\) because \(a^0 = 1\).
• Asymptote: The graph approaches the \(x\)-axis (\(y=0\)) but never touches it. The \(x\)-axis is a horizontal asymptote.

Analogy: Think of a rumor spreading. Initially, it spreads slowly (small \(x\)). Then, it reaches many people quickly (steep curve). This sudden, rapid increase is the hallmark of exponential growth.

3. Introducing Logarithms (The Inverse Function)

Logarithms are simply a different way of writing an exponential relationship. They answer the question: "What power do I raise the base to, to get this number?"

The Definition of a Logarithm

The exponential statement and the logarithmic statement are two sides of the same coin:

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

In English: "The power \(y\) that you raise the base \(a\) to, is \(x\)."

Key terms:

• \(a\) is the base (it must be positive, \(a > 0\), and \(a \neq 1\)).
• \(x\) is the argument (the number you are taking the log of, must be positive, \(x > 0\)).
• \(y\) is the exponent (the power).

Example Translations

• If \(2^3 = 8\), then \(\log_2 8 = 3\). (The power we raise 2 to, to get 8, is 3.)
• If \(10^{-2} = 0.01\), then \(\log_{10} 0.01 = -2\).

Crucial Insight: The function \(y = \log_a x\) is the inverse of \(y = a^x\). This means their graphs are reflections of each other across the line \(y=x\).

Because \(a^x\) has an asymptote at \(y=0\), \(\log_a x\) has an asymptote at \(x=0\) (the \(y\)-axis). The domain of \(\log_a x\) is \(x > 0\). You cannot take the logarithm of a negative number or zero.

4. The Laws of Logarithms (The Tools)

The power of logarithms comes from their laws, which allow us to combine and separate logarithmic expressions. These laws mirror the index laws.

Law 1: The Product Law (Addition)

The log of a product is the sum of the logs.

\(\log_a (xy) = \log_a x + \log_a y\)

Memory Aid: If you are multiplying arguments, you add the powers (logs).

Law 2: The Quotient Law (Subtraction)

The log of a quotient (division) is the difference of the logs.

\(\log_a (\frac{x}{y}) = \log_a x - \log_a y\)

Memory Aid: If you are dividing arguments, you subtract the powers (logs).

Law 3: The Power Law (Multiplication)

The exponent (power) inside a logarithm can be brought down to the front as a multiplier.

\(\log_a (x^k) = k \log_a x\)

This is the most powerful law for solving equations, as it lets us move the variable \(k\) out of the exponent!

Law 4: The Change of Base Rule

Sometimes your calculator only has log base 10 (denoted as \(\log x\) or \(\log_{10} x\)) and natural log (\(\ln x\)). If you encounter a log with an unusual base, you must use the change of base rule.

\(\log_a x = \frac{\log_b x}{\log_b a}\)

Usually, we choose base \(b=10\) or \(b=e\).

Step-by-Step Example (Simplifying)

Simplify: \(2 \log_3 6 - \log_3 4\)

1. Use Power Law (Law 3): Bring the coefficient (2) up as a power.
   \(\log_3 (6^2) - \log_3 4 = \log_3 36 - \log_3 4\)
2. Use Quotient Law (Law 2): Subtraction becomes division.
   \(\log_3 (\frac{36}{4}) = \log_3 9\)
3. Evaluate: Ask: "What power do I raise 3 to, to get 9?"
   \(\log_3 9 = 2\) (Since \(3^2 = 9\))

5. Solving Exponential and Logarithmic Equations

This is where we put the laws into action. The main goal is usually to isolate the variable, \(x\).

Case A: Solving Exponential Equations (\(a^x = b\))

When the variable is in the power, you must introduce logarithms.

Step-by-Step Process (Solving \(5^x = 120\)):

1. Take logs of both sides: Choose any convenient base (usually base 10 or base \(e\)).
   \(\log (5^x) = \log (120)\)
2. Use the Power Law (Law 3): Move the variable \(x\) to the front.
   \(x \log 5 = \log 120\)
3. Isolate \(x\): Divide by \(\log 5\).
   \(x = \frac{\log 120}{\log 5}\)
4. Calculate: Use your calculator.
   \(x \approx 2.97\) (3 s.f.)

Common Mistake to Avoid: \(\frac{\log 120}{\log 5} \neq \log (\frac{120}{5})\). Remember, the quotient law only applies if you have subtraction, not division of two separate logs!

Case B: Solving Logarithmic Equations

When solving equations involving only logarithms, try to condense the equation until you have a single log on one side.

Step-by-Step Process (Solving \(\log_2 (x+2) + \log_2 x = 3\)):

1. Condense using Product Law: Addition becomes multiplication.
   \(\log_2 ( (x+2)x ) = 3\)
2. Convert to Exponential Form: Use the definition \(\log_a x = y \iff a^y = x\).
   \(2^3 = x(x+2)\)
3. Simplify and Solve the Quadratic:
   \(8 = x^2 + 2x\)
   \(x^2 + 2x - 8 = 0\)
   \((x+4)(x-2) = 0\)
4. Check Solutions: We get \(x = -4\) or \(x = 2\). Crucially, the argument of a log must be positive.
   • If \(x=-4\), \(\log_2 (-4)\) is undefined. So, reject \(x=-4\).
   • If \(x=2\), \(\log_2 2\) and \(\log_2 4\) are valid. So, \(x=2\) is the only solution.

6. The Natural Base \(e\) and Natural Logarithms \(\ln x\)

In mathematics, physics, and economics, one base appears so frequently that it is called the natural base. This is the number \(e\), often called Euler’s number.

Introducing \(e\)

The number \(e\) is an irrational number, approximately \(2.71828\). It arises naturally in processes of continuous growth.

Did you know? \(e\) is defined as the limit of compound interest when interest is compounded continuously. If you invest $1 at 100% interest for 1 year, the maximum amount you can achieve by compounding continuously is exactly \(\$e\).

The function \(y = e^x\) is the natural exponential function.

Introducing \(\ln x\)

The logarithm with base \(e\) is called the natural logarithm.

\(\log_e x\) is written as \(\ln x\)

Since \(\ln x\) is just a logarithm, all the logarithm laws (Product, Quotient, Power) apply exactly the same way to natural logs.

Key Identities for \(e\) and \(\ln\):

• \(e^{\ln x} = x\)
• \(\ln (e^x) = x\)
• \(\ln e = 1\) (Since \(\log_e e = 1\))
• \(\ln 1 = 0\) (Since \(\log_e 1 = 0\))

7. Modeling with Exponentials (Real-World Applications)

Exponential functions are vital for modeling real-world phenomena. You will often encounter models involving the base \(e\).

The Standard Growth/Decay Model

\(P = A e^{kt}\)

Where:

• \(P\) is the quantity (e.g., population, amount of substance) at time \(t\).
• \(A\) is the initial quantity (when \(t=0\)).
• \(t\) is the time.
• \(k\) is the constant of proportionality (the growth/decay rate).

If \(k > 0\), it's growth. If \(k < 0\), it's decay.

Linearizing Data (The \(Y = mX + c\) approach)

Sometimes, experimental data might follow an exponential model like \(y = A x^k\) or \(y = A e^{kx}\). By taking logarithms, we can transform these curves into straight lines, making it easier to find the constants \(A\) and \(k\) using linear regression (which you may study in Statistics, or by finding the gradient/intercept).

Example: Transforming \(y = A e^{kx}\)

1. Take natural logs of both sides:
   \(\ln y = \ln (A e^{kx})\)
2. Use the Product Law:
   \(\ln y = \ln A + \ln (e^{kx})\)
3. Use the Log/Exponent inverse property (\(\ln e^{kx} = kx\)):
   \(\ln y = \ln A + kx\)

We now have an equation in the form \(Y = mX + c\):

• \(Y = \ln y\)
• \(X = x\)
• The gradient \(m = k\)
• The \(Y\)-intercept \(c = \ln A\)

If you plot \(\ln y\) against \(x\), you will get a straight line!

8. Differentiation of Exponential and Logarithmic Functions

This section introduces how to find the derivative (\(\frac{dy}{dx}\)) of the natural exponential and logarithmic functions. You will find that differentiating \(e^x\) is surprisingly simple!

Rule 1: Differentiating \(e^{kx}\)

If \(y = e^{kx}\), then \(\frac{dy}{dx} = k e^{kx}\)

(The derivative of \(e^x\) is simply \(e^x\). For \(e^{kx}\), we multiply by the derivative of the power, which is \(k\).)

Examples:

• If \(y = e^{5x}\), then \(\frac{dy}{dx} = 5e^{5x}\)
• If \(y = 3e^{-2x}\), then \(\frac{dy}{dx} = 3 \times (-2) e^{-2x} = -6e^{-2x}\)

Rule 2: Differentiating \(\ln (ax+b)\)

If \(y = \ln x\), then \(\frac{dy}{dx} = \frac{1}{x}\)

For a more complex argument:

If \(y = \ln (ax+b)\), then \(\frac{dy}{dx} = \frac{a}{ax+b}\)

Analogy: The derivative of \(\ln(\text{stuff})\) is always: \(\frac{\text{derivative of the stuff}}{\text{the stuff}}\)

Examples:

• If \(y = \ln (x)\), then \(\frac{dy}{dx} = \frac{1}{x}\)
• If \(y = \ln (5x-3)\), then \(\frac{dy}{dx} = \frac{5}{5x-3}\)

Key Takeaway for Differentiation: These are standard results that you must memorize. The simplicity of the derivative of \(e^x\) is why the base \(e\) is called the natural base!