Welcome to Exponentials and Logarithms!
Hello there! This chapter is incredibly important. Exponentials and logarithms describe some of the most fundamental processes in science, finance, and the real world—think population growth, radioactive decay, or compound interest.
Don't worry if the symbols seem intimidating at first. We will break down these concepts step-by-step, linking them back to familiar rules you already know: the laws of indices. Logs are just a special way of asking: "What is the power?"
Let’s dive in and master these powerful functions!
1. The Exponential Function \(y = a^x\)
An exponential function is where the variable (\(x\)) is in the power (the exponent).
What is an Exponential Function?
The general form of an exponential function is:
\[y = a^x \]
Where:
- \(a\) is the base (a constant, where \(a > 0\) and \(a \neq 1\)).
- \(x\) is the variable exponent.
Example: \(y = 2^x\), \(y = 10^x\), or the very special \(y = e^x\).
Sketching the Graph of \(y = a^x\)
No matter the base \(a\) (as long as \(a>1\)), the graphs share key features:
- The graph always passes through the point (0, 1), because \(a^0 = 1\).
- The x-axis (\(y=0\)) is a horizontal asymptote. The curve gets closer and closer to \(y=0\) but never touches it.
- If \(a > 1\), the function shows exponential growth (it increases rapidly).
Quick Review: Key feature for all \(y=a^x\): passes through (0, 1), asymptote \(y=0\).
The Natural Exponential Function: \(y = e^x\)
In A-Level Maths, we often use a specific base called \(e\). It is an irrational number, approximately:
\[e \approx 2.71828 \]
The function \(y = e^x\) is called the Natural Exponential Function. It is the most important exponential function in calculus because of its unique differentiation property (which we'll see later!).
Modelling Exponential Growth and Decay (P2 Content)
Exponential functions are used to model situations where the rate of change is proportional to the current amount.
The standard model is:
\[P = A e^{kt} \]
- \(A\) is the initial amount (when \(t=0\)).
- \(t\) is time.
- \(k\) is the growth/decay constant.
1. Exponential Growth: If \(k\) is positive (\(k > 0\)). (e.g., population increase, continuously compounding interest).
2. Exponential Decay: If \(k\) is negative (\(k < 0\)). (e.g., radioactive decay, cooling of an object).
Did you know? The value \(e\) is often called Euler's number. It naturally arises when calculating growth that is compounded continuously, rather than annually or monthly.
2. Review of the Laws of Indices
Before diving into logarithms, we must be fluent in index laws, as the log laws are just these rules translated!
Summary of Index Laws (for all rational exponents)
Let \(a\), \(m\), and \(n\) be numbers:
- Multiplication Rule: When multiplying powers with the same base, add the exponents.
\[ a^m \times a^n = a^{m+n} \] - Division Rule: When dividing powers with the same base, subtract the exponents.
\[ a^m \div a^n = a^{m-n} \] - Power of a Power Rule: When raising a power to another power, multiply the exponents.
\[ (a^m)^n = a^{mn} \]
Other crucial rules:
- \[ a^1 = a \]
- \[ a^0 = 1 \]
- \[ a^{-n} = \frac{1}{a^n} \]
- \[ a^{1/n} = \sqrt[n]{a} \]
Key Takeaway: Indices allow us to work with powers directly. Logarithms are what we use when we want to find the power itself!
3. Logarithms: The Inverse Function
The Definition of a Logarithm
A logarithm is simply the inverse operation of exponentiation.
Analogy: If \(2^x = 8\), you ask, "2 to what power equals 8?" The answer is \(x=3\). The logarithm is the mathematical tool that asks this question.
The key equivalence you must learn and use fluently is:
\[ y = a^x \quad \iff \quad x = \log_a y \]
- \(a\) is the base (must be positive and \(a \neq 1\)).
- \(x\) is the exponent (the logarithm).
Example: \(100 = 10^2 \iff 2 = \log_{10} 100\).
Common Mistake to Avoid: You cannot take the logarithm of a negative number or zero! The domain of \(\log_a x\) is \(x>0\).
Special Logarithms
Just as \(e\) is the special base for exponentials, logarithms based on \(e\) have a special name:
The Natural Logarithm (\(\ln x\))
This is the logarithm to the base \(e\). It is so common it has its own notation:
\[ \ln x \quad \text{means} \quad \log_e x \]
When you see \(\ln x\) or the 'ln' button on your calculator, think base \(e\).
Logarithms to Base 10 (\(\log_{10} x\))
Sometimes written simply as \(\log x\). This is often used in science but is less common than \(\ln x\) in A-Level Pure Mathematics.
The Graph of \(y = \ln x\)
Since \(y = \ln x\) is the inverse function of \(y = e^x\), its graph is a reflection of \(y = e^x\) in the line \(y = x\).
- The graph passes through (1, 0) (since \(\ln 1 = 0\)).
- The y-axis (\(x=0\)) is a vertical asymptote.
- The domain is \(x > 0\). The range is all real numbers (\(y \in \mathbb{R}\)).
4. The Laws of Logarithms
These laws allow you to manipulate, combine, and split logarithms, which is essential for solving equations. They directly mirror the laws of indices!
Law 1: The Addition Law (Product Rule)
When you add logs of the same base, you multiply the arguments.
\[ \log_a x + \log_a y = \log_a (xy) \]
Memory Aid: Logs turn multiplication (indices law 1) into addition.
Law 2: The Subtraction Law (Quotient Rule)
When you subtract logs of the same base, you divide the arguments.
\[ \log_a x - \log_a y = \log_a \left(\frac{x}{y}\right) \]
Memory Aid: Logs turn division (indices law 2) into subtraction.
Law 3: The Power Law
The power inside the log can be moved to the front as a multiplier (coefficient).
\[ k \log_a x = \log_a (x^k) \]
This is the most powerful law for solving equations, as it allows you to bring down the exponent.
Other Useful Identities
- Log of the Base: \(\log_a a = 1\) (Because \(a^1 = a\)).
- Log of One: \(\log_a 1 = 0\) (Because \(a^0 = 1\)).
Quick Review: Index vs. Log Laws
Indices: \(a^m \times a^n = a^{m+n}\)
Logs: \(\log(m) + \log(n) = \log(mn)\)
Notice how multiplication becomes addition in the world of logs?
5. Solving Equations Involving Exponentials and Logarithms
The primary skill here is knowing when to switch between exponential and logarithmic form, and using the Power Law to handle variables in the exponent.
Type 1: Solving Exponential Equations (\(a^x = b\))
If the variable is in the exponent, you must use logarithms to solve it.
Step-by-Step Example: Solve \(3^{2x} = 2\)
- Take logs of both sides. Since we will use a calculator, it is standard practice to use the natural logarithm, \(\ln\).
\[ \ln(3^{2x}) = \ln(2) \] - Use the Power Law (Law 3). Bring the exponent down.
\[ 2x \ln(3) = \ln(2) \] - Isolate \(x\). Divide by the coefficient \(2 \ln(3)\).
\[ x = \frac{\ln(2)}{2 \ln(3)} \] - Calculate the value. (Using a calculator).
Encouragement: Don't worry if the answers look messy! Leaving your answer in the form \(\frac{\ln 2}{2 \ln 3}\) is often preferred for exactness unless the question specifies rounding.
Type 2: Solving Logarithmic Equations
If the equation contains multiple log terms, use the log laws to condense them into a single log term, then convert to exponential form.
Step-by-Step Example: Solve \(\log_2(x+1) + \log_2(x-1) = 3\)
- Condense the left side using the Addition Law.
\[ \log_2((x+1)(x-1)) = 3 \] \[ \log_2(x^2 - 1) = 3 \] - Convert to exponential form. Remember \(\log_a y = x \iff a^x = y\).
\[ x^2 - 1 = 2^3 \] \[ x^2 - 1 = 8 \] - Solve the resulting equation.
\[ x^2 = 9 \] \[ x = 3 \quad \text{or} \quad x = -3 \] - Check the Domain! Remember, the argument of a log must be positive. If \(x=-3\), then \(\log_2(x-1) = \log_2(-4)\), which is invalid.
Solution: \(x=3\) is the only valid answer.
Common Mistake to Avoid: ALWAYS check your solutions against the original equation to ensure you are not taking the log of a negative number.
6. Calculus of Exponentials and Logarithms (A-Level P2 Content)
The natural base \(e\) is crucial because it makes differentiation and integration remarkably simple.
Differentiation
1. Differentiating \(e^x\)
The derivative of \(e^x\) is \(e^x\). This is what makes \(e\) the "natural" base—it is its own derivative!
\[ \frac{d}{dx} (e^x) = e^x \]
If you have a constant multiplier in the power (using the Chain Rule):
\[ \frac{d}{dx} (e^{kx}) = k e^{kx} \]
2. Differentiating \(\ln x\)
The derivative of the natural logarithm \(\ln x\) is \(\frac{1}{x}\).
\[ \frac{d}{dx} (\ln x) = \frac{1}{x} \quad \text{for } x>0 \]
If you have a function inside the log (using the Chain Rule):
\[ \frac{d}{dx} (\ln(f(x))) = \frac{f'(x)}{f(x)} \]
Example: If \(y = \ln(x^2+3)\), then \(y' = \frac{2x}{x^2+3}\).
Integration (The Reverse Process)
Since integration is the reverse of differentiation, these rules follow directly:
1. Integrating \(e^x\)
\[ \int e^x \, dx = e^x + c \]
And using reversal of the Chain Rule (integration by inspection):
\[ \int e^{kx} \, dx = \frac{1}{k} e^{kx} + c \]
2. Integrating \(\frac{1}{x}\)
This is arguably the most important new integration rule in P2, as it covers the gap left by the power rule of integration (\(\int x^n dx = \frac{x^{n+1}}{n+1}\)), which doesn't work when \(n = -1\).
\[ \int \frac{1}{x} \, dx = \ln|x| + c \]
CRUCIAL POINT: The Absolute Value (\(|x|\))
We must use \(\ln|x|\) because the integral is valid for \(x \neq 0\). Although \(\ln x\) is only defined for \(x>0\), the graph of \(y=1/x\) exists for negative \(x\). The absolute value ensures the argument of the logarithm is always positive, satisfying the domain restriction.
3. Integrating Ratios (Inspection/Substitution)
If you have a fraction where the numerator is the derivative of the denominator, the integral is the natural log of the denominator.
\[ \int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + c \]
Example: \(\int \frac{2x}{x^2 + 5} \, dx\). Here, \(f(x) = x^2+5\) and \(f'(x) = 2x\).
Solution: \(\ln|x^2 + 5| + c\).
Key Takeaway: The functions \(e^x\) and \(\ln x\) are fundamental to calculus. Remember their differentiation and integration rules—especially \(\int \frac{1}{x} \, dx = \ln|x| + c\).