Mathematics - Additional (0606) | Logarithmic and exponential functions

Chapter 6: Logarithmic and Exponential Functions

Hello there! Welcome to one of the most powerful and exciting topics in Additional Mathematics: Logarithms and Exponential Functions.

Don't worry if these terms sound complicated. They are just the mathematical tools we use to deal with things that grow or shrink incredibly fast, like population growth, radioactive decay, or compound interest. They are inverses of each other—think of them as two sides of the same coin!

In this chapter, we will learn their fundamental properties, how to draw their graphs, and, most importantly, how to use them to solve tricky equations. Let's get started!

6.1 The Natural Exponential Function: \(y = e^x\)

The exponential function is a function where the variable is in the index (exponent). While you are familiar with bases like 2 or 10, Additional Mathematics often focuses on a very special base: the number $e$.

What is $e$?

$e$ is the Natural Base. It's an irrational number, just like \(\pi\), and its value is approximately 2.71828. It appears naturally in processes involving continuous growth.

The Natural Exponential Function is written as \(f(x) = e^x\).

Properties of the Graph \(y = e^x\)

• Y-intercept: When \(x=0\), \(y = e^0 = 1\). The graph always passes through \((0, 1)\).
• Domain and Range: The domain (x-values) is all real numbers. The range (y-values) is \(y > 0\).
• Asymptote: The graph approaches the x-axis (\(y=0\)) but never touches it. This line is called the horizontal asymptote.
• Growth: Since \(e > 1\), the function always increases as \(x\) increases (it's an increasing function).

Did you know? The number $e$ is sometimes called Euler's number (after the mathematician Leonhard Euler) and is essential in finance and calculus because its derivative is itself!

Graph Transformations (Syllabus Scope)

You must be able to sketch and understand graphs of the form:

$$y = k e^{nx} + a$$

• \(a\): Vertical Shift (Determines the Asymptote)
  This shifts the entire graph up or down. The horizontal asymptote is always \(y = a\).
• \(k\): Vertical Stretch/Compression
  This stretches the graph vertically.
• \(n\): Horizontal Scaling
  If \(n\) is large, the growth is faster.

Example: For \(y = 3e^{2x} + 5\), the horizontal asymptote is \(y = 5\). The graph will always be above the line \(y=5\).

6.2 The Natural Logarithmic Function: \(y = \ln x\)

The logarithmic function is the inverse of the exponential function. It answers the question: "What exponent do I need to raise the base to, to get this number?"

The Definition of a Logarithm

In general, if \(b^y = x\), then \(\log_b x = y\).

When the base \(b\) is the natural number \(e\), we use a special notation: $\ln x$ (read as "lon x" or "natural log of x").

Therefore, \(\ln x\) means \(\log_e x\).

The Inverse Relationship (The "Undo" Button)

Since \(e^x\) and \(\ln x\) are inverse functions, they cancel each other out:

• \(\ln(e^x) = x\)
• \(e^{(\ln x)} = x\)

Analogy: If you put on your socks (\(e^x\)) and then take them off (\(\ln x\)), you end up where you started (x).

Properties of the Graph \(y = \ln x\)

The graph of \(y = \ln x\) is a reflection of \(y = e^x\) in the line \(y = x\).

• X-intercept: When \(y=0\), \(\ln x = 0\), so \(x = e^0 = 1\). The graph always passes through \((1, 0)\).
• Domain: Logarithms are only defined for positive numbers. You cannot take the log of zero or a negative number. Thus, the domain is \(x > 0\).
• Asymptote: The graph approaches the y-axis (\(x=0\)) but never touches it. This is the vertical asymptote.
• Range: The range (y-values) is all real numbers.

Graph Transformations (Syllabus Scope)

You must be able to sketch and understand graphs of the form:

$$y = k \ln(ax + b)$$

The most important part here is determining the vertical asymptote.

The argument of the logarithm (\(ax + b\)) must be greater than zero. The asymptote occurs where the argument equals zero:

$$ax + b = 0 \quad \Rightarrow \quad x = -\frac{b}{a}$$

Example: For \(y = 2 \ln(x - 3)\), the domain requires \(x - 3 > 0\), so \(x > 3\). The vertical asymptote is \(x = 3\).

6.3 The Laws of Logarithms (All Bases)

These three laws are your best friends when simplifying logarithmic expressions or solving equations. They work for any base ($b$), including $e$ (\(\ln\)) and 10 (\(\lg\)).

Law 1: The Product Law (Multiplication becomes Addition)

\(\log_b (XY) = \log_b X + \log_b Y\)

Example: \(\ln(4x) = \ln 4 + \ln x\)

Law 2: The Quotient Law (Division becomes Subtraction)

\(\log_b \left(\frac{X}{Y}\right) = \log_b X - \log_b Y\)

Example: \(\lg \left(\frac{100}{y}\right) = \lg 100 - \lg y = 2 - \lg y\)

Law 3: The Power Law (Exponents come down)

\(\log_b (X^p) = p \log_b X\)

This is the most crucial law for solving equations because it allows us to bring the variable down from the exponent!

Example: \(\ln(x^3) = 3 \ln x\)

Special Properties

• \(\log_b 1 = 0\) (Because \(b^0 = 1\) for any base $b$)
• \(\log_b b = 1\) (Because \(b^1 = b\))
• \(\log_{10} x\) is often written as \(\lg x\).

The Change of Base Rule (Essential for Calculations)

Sometimes you are given a logarithm in a strange base, say base 5, but your calculator only handles base 10 (\(\lg\)) or base $e$ (\(\ln\)). The Change of Base rule allows you to convert it:

\(\log_a b = \frac{\log_c b}{\log_c a}\)

We usually change to base $e$ (natural log) or base 10:

\(\log_a b = \frac{\ln b}{\ln a} = \frac{\lg b}{\lg a}\)

Example: To calculate \(\log_2 15\), we write \(\frac{\ln 15}{\ln 2}\) or \(\frac{\lg 15}{\lg 2}\).

6.4 Solving Equations Involving Logarithms and Exponentials

The key to solving these equations is knowing when to use logs and when to use exponentials, often switching between the two forms.

Case 1: Solving Exponential Equations (\(a^x = b\))

If the variable you are solving for is in the exponent, you must use logarithms to bring it down.

Step-by-Step: Solving \(5^x = 30\)

1. Take the log of both sides. Use the natural log (\(\ln\)) as it's efficient:
   \(\ln(5^x) = \ln(30)\)
2. Apply the Power Law (Law 3): Bring \(x\) down as a multiplier:
   \(x \ln 5 = \ln 30\)
3. Isolate \(x\):
   \(x = \frac{\ln 30}{\ln 5}\)
4. Calculate (if required): Use your calculator to find the numerical answer (ensure you give sufficient significant figures).

Remember: You can take the log to any base, but using \(\ln\) or \(\lg\) simplifies the calculation on Paper 2.

Case 2: Solving Logarithmic Equations

If the variable is trapped inside a logarithm, you must use exponentiation (the inverse) to release it.

Step-by-Step: Solving \(\ln(2x - 1) = 4\)

1. Isolate the logarithm. (It's already isolated here.)
2. Convert to Exponential Form: Since \(\ln\) has base \(e\), raise both sides as powers of \(e\):
   \(e^{\ln(2x - 1)} = e^4\)
3. Simplify: The \(e\) and \(\ln\) cancel out:
   \(2x - 1 = e^4\)
4. Solve for \(x\):
   \(2x = e^4 + 1\)
   \(x = \frac{e^4 + 1}{2}\)

Case 3: Using Substitution to Solve Log/Exp Equations

Sometimes, equations look complicated but are actually quadratic in disguise.

Example: Solve \(2(e^x)^2 + 3e^x - 2 = 0\).

1. Let \(y\) equal the term being repeated. Let \(y = e^x\).
2. Substitute and Solve the Quadratic:
   \(2y^2 + 3y - 2 = 0\)
   \((2y - 1)(y + 2) = 0\)
   So, \(y = \frac{1}{2}\) or \(y = -2\).
3. Substitute back and Solve for \(x\):
   Case 1: \(e^x = \frac{1}{2}\). Take \(\ln\) of both sides: \(x = \ln(0.5)\).
   Case 2: \(e^x = -2\). Stop! Since \(e^x\) must always be positive (\(e^x > 0\)), this solution is impossible or invalid.

The final solution is \(x = \ln(0.5)\).

You've now covered the essential tools for logarithmic and exponential functions! Practice these properties and laws, and you will find these questions manageable. Keep practicing, you got this!