Functions (Grade 12 NSC Matric Mathematics): Revision Notes

Exponential Functions & Logarithms

What are exponential functions?

An exponential function is a mathematical function where a constant base is raised to a variable power. The general form is:

$f(x) = b^x$

where:

• $b$ is the base (a positive constant)
• $x$ is the exponent or index (the variable)
• $b > 0$ and $b ≠ 1$

Understanding exponents

An exponent tells us how many times to multiply the base by itself. For example:

• $10^3 = 10 × 10 × 10 = 1000$
• The base is 10, the exponent is 3

Behaviour of exponential functions

The value of the base determines how the exponential function behaves:

When $b > 1$ (exponential growth)

• The function is increasing
• The graph rises steeply from left to right
• As $x$ increases, $f(x)$ grows rapidly
• As $x$ decreases, $f(x)$ approaches zero

When $0 < b < 1$ (exponential decay)

• The function is decreasing
• The graph falls from left to right
• As $x$ increases, $f(x)$ approaches zero
• As $x$ decreases, $f(x)$ grows rapidly

Finding inverse functions

When we want to find the inverse of an exponential function, we face a problem. Consider $y = b^x$:

To find the inverse, we interchange $x$ and $y$: $x = b^y$

However, we cannot easily solve for $y$ when it appears as an exponent. This is where logarithms come to our rescue.

Introduction to logarithms

Definition: If $x = b^y$, then $y = \log_b(x)$, where $b > 0$, $b ≠ 1$ and $x > 0$.

This means that logarithms are the inverse functions of exponentials. The logarithm answers the question: "To what power must I raise the base $b$ to get $x$?"

Restrictions on logarithms

Converting between forms

To find the inverse of $y = b^x$:

1. Interchange $x$ and $y$: $x = b^y$
2. Make $y$ the subject: $y = \log_b x$

Therefore, if $f(x) = b^x$, then $f^{-1}(x) = \log_b x$.

Types of logarithms

Common logarithm

The common logarithm has base 10 and can be written as $\log_{10} x = \log x$. When no base is shown, we assume base 10.

Natural logarithm

The natural logarithm has base $e$ (approximately 2.71) and can be written as $\log_e x = \ln x$.

Special logarithmic values

Laws of logarithms

These laws help us simplify logarithmic expressions:

Power law

$\log_a(x^b) = b \log_a x \text{ (where } x > 0\text{)}$

Change of base formula

$\log_a x = \frac{\log_b x}{\log_b a} \text{ (where } b > 0 \text{ and } b ≠ 1\text{)}$

Special applications of change of base

1. $\log_a x = \frac{1}{\log_x a}$
2. $\log_a(1/x) = \log_a x^{-1} = -\log_a x$

Using calculators for logarithms

Finding logarithmic values

Most scientific calculators have three logarithm buttons:

• log: for common logarithms (base 10)
• ln: for natural logarithms (base e)
• log with base: for logarithms with any base

Graphs of logarithmic functions

Basic logarithmic function properties

For $f(x) = \log_b x$ where $b > 0$ and $b ≠ 1$:

Relationship between exponential and logarithmic graphs

Exponential and logarithmic functions are inverses of each other. This means:

• The domain of one equals the range of the other
• Their graphs are reflections of each other about the line $y = x$
• If $(a, b)$ is on the exponential graph, then $(b, a)$ is on the logarithmic graph

Applications of logarithms

Logarithms have many practical applications in real-world situations:

Population growth

Formula: $A = P(1 + i)^n$

Where $A$ is final amount, $P$ is initial amount, $i$ is growth rate, and $n$ is time period.

Other applications include:

• Seismology: calculating earthquake magnitudes
• Finance: compound interest calculations
• Science: radioactive decay rates
• Biology: population growth rates
• Chemistry: pH levels

Key graph summary

For exponential functions $y = b^x$ and logarithmic functions $y = \log_b x$:

• When $b > 1$: exponential shows growth, logarithmic increases slowly
• When $0 < b < 1$: exponential shows decay, logarithmic decreases
• Both functions pass through key points that are inverses of each other
• The line $y = x$ serves as the axis of symmetry between inverse functions