Logarithms (VCE SSCE Mathematical Methods): Revision Notes

Logarithms

Introduction to logarithms

A logarithm provides an alternative way to express an exponential relationship. Let's examine the example:

$$2^3 = 8$$

We can rewrite this same relationship using logarithms as:

$$\log_2 8 = 3$$

This is read as "the logarithm of 8 to the base 2 is equal to 3".

Definition

For $a \in \mathbb{R}^+ \backslash \{1\}$, the logarithm function with base $a$ is defined as follows:

$$a^x = y \text{ is equivalent to } \log_a y = x$$

Converting between forms

Here are some examples showing how to convert between exponential and logarithmic forms:

• $3^2 = 9$ is equivalent to $\log_3 9 = 2$

• $10^4 = 10000$ is equivalent to $\log_{10} 10000 = 4$

• $a^0 = 1$ is equivalent to $\log_a 1 = 0$

Evaluating logarithms

To find the value of a logarithm, we ask ourselves a key question: "What power of the base gives this number?"

For example:

• To find $\log_2 32$, we ask "What power of 2 gives 32?"
• To find $\log_3 81$, we ask "What power of 3 gives 81?"

Logarithms as inverse functions

For each base $a \in \mathbb{R}^+ \backslash \{1\}$, the exponential function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = a^x$ is one-to-one and therefore has an inverse function.

The inverse of the exponential function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = a^x$ is the logarithmic function $f^{-1}: \mathbb{R}^+ \to \mathbb{R}$, $f^{-1}(x) = \log_a x$.

Key inverse properties

Graphical representation

Since exponential and logarithmic functions are inverses, their graphs are reflections of each other in the line $y = x$.

The left graph shows the case where $a > 1$, and the right graph shows the case where $0 < a < 1$. Notice how the exponential curve and logarithmic curve are mirror images across the line $y = x$.

Natural logarithm

Earlier we defined the number $e$ and the important function $f(x) = e^x$. The inverse of this function is $f^{-1}(x) = \log_e x$.

Because the logarithm function with base $e$ is known as the natural logarithm, the expression $\log_e x$ is also written as ln x.

This notation is commonly used in mathematics and science, so you should be familiar with both forms:

$$\log_e x = \ln x$$

Common logarithm

The function $f(x) = \log_{10} x$ has both historical and practical importance. Before calculators were widely available, logarithms were used as a calculating device, and it was often base 10 that was used. By simplifying calculations, logarithms contributed to the advancement of science, especially astronomy. In schools, books of tables of logarithms were provided for calculations up to the 1970s.

You can understand the practicality of base 10 by observing:

• $\log_{10} 10 = 1$

• $\log_{10} 100 = 2$

• $\log_{10} 1000 = 3$

• $\log_{10} 10000 = 4$

• $\log_{10} 0.1 = -1$

• $\log_{10} 0.01 = -2$

• $\log_{10} 0.001 = -3$

• $\log_{10} 0.0001 = -4$

This pattern makes base 10 logarithms particularly useful for expressing very large or very small numbers in a compact form.

Laws of logarithms

The index laws are used to establish rules for computations with logarithms. These laws are essential for simplifying logarithmic expressions and solving logarithmic equations.

Law 1: Logarithm of a product

The logarithm of a product equals the sum of the logarithms:

$$\log_a(mn) = \log_a m + \log_a n$$

Proof: Let $\log_a m = x$ and $\log_a n = y$, where $m$ and $n$ are positive real numbers.

Then $a^x = m$ and $a^y = n$, and therefore:

$$mn = a^x \times a^y = a^{x+y}$$

(using the first index law)

Hence $\log_a(mn) = x + y = \log_a m + \log_a n$

Example:

$$\log_{10} 200 + \log_{10} 5 = \log_{10}(200 \times 5)$$ $$= \log_{10} 1000 = 3$$

Law 2: Logarithm of a quotient

The logarithm of a quotient equals the difference of the logarithms:

$$\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n$$

Proof: Let $\log_a m = x$ and $\log_a n = y$, where $m$ and $n$ are positive real numbers.

Then as before $a^x = m$ and $a^y = n$, and therefore:

$$\frac{m}{n} = \frac{a^x}{a^y} = a^{x-y}$$

(using the second index law)

Hence $\log_a\left(\frac{m}{n}\right) = x - y = \log_a m - \log_a n$

Example:

$$\log_2 32 - \log_2 8 = \log_2\left(\frac{32}{8}\right)$$ $$= \log_2 4 = 2$$

Law 3: Logarithm of a power

$$\log_a(m^p) = p \log_a m$$

Proof: Let $\log_a m = x$. Then $a^x = m$, and therefore:

$$m^p = (a^x)^p = a^{xp}$$

(using the third index law)

Hence $\log_a(m^p) = xp = p \log_a m$

Example:

$$\log_2 32 = \log_2(2^5) = 5$$

Law 4: Logarithm of a reciprocal

$$\log_a(m^{-1}) = -\log_a m$$

Proof: This follows from Law 3 with $p = -1$.

Example:

$$\log_a\left(\frac{1}{2}\right) = \log_a(2^{-1}) = -\log_a 2$$

Law 5: Special values

$$\log_a 1 = 0 \text{ and } \log_a a = 1$$

Proof: Since $a^0 = 1$, we have $\log_a 1 = 0$

Since $a^1 = a$, we have $\log_a a = 1$

Solving logarithmic equations

When solving logarithmic equations, we use the laws of logarithms and the relationship between logarithmic and exponential forms.