Apply indices and logarithms is a fundamental skill for applied biology professionals in Kenya, particularly when dealing with exponential growth of populations, decay of substances, and pH calculations in environmental and medical contexts. This chapter equips students with the ability to manipulate powers and solve equations involving indices, which is essential for modeling biological processes such as bacterial growth or radioactive decay of samples. Mastery of indices also supports understanding logarithmic scales used in enzyme kinetics and sound intensity measurements.

4.1 Indices

Indices represent the power to which a number, called the base, is raised. In applied biology, indices are used to simplify expressions involving repeated multiplication, such as growth rates or dilution factors. This section covers the concept of bases, the fundamental laws governing indices, and solving indicial equations.

4.1.1 Bases

Indices notation expresses repeated multiplication of the same number, the base, raised to an exponent, the index. The base can be any real number, and the index indicates how many times the base multiplies itself.

The general form is:

$$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$$

where \(a\) is the base and \(n\) the index (a positive integer).

Worked Examples

Example 1: Calculate \(3^4\) representing the number of cells after 4 replication cycles if each cell divides into 3.

Given: base \(a = 3\), index \(n = 4\)

$$a^n = 3^4$$

$$= 3 \times 3 \times 3 \times 3$$

$$= 81$$

Answer: 81 cells

Example 2: Find the value of \(5^3\) representing the concentration factor after three successive dilutions by factor 5.

Given: \(a = 5\), \(n = 3\)

$$5^3 = 5 \times 5 \times 5$$

$$= 125$$

Answer: 125 times concentration

Example 3: Compute \(2^6\) for a bacterial population doubling every hour over 6 hours.

Given: \(a = 2\), \(n = 6\)

$$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$$

$$= 64$$

Answer: 64 bacteria

4.1.2 Laws of Indices

The laws of indices allow simplification and manipulation of expressions involving powers with the same base. These laws are essential in applied biology for calculating growth or decay over multiple stages or combining rates.

The main laws are:

1. Product Rule: \(a^m \times a^n = a^{m+n}\)
2. Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) where \(m > n\)
3. Power of a Power: \((a^m)^n = a^{m \times n}\)

Worked Examples

Example 1: Calculate \(2^3 \times 2^4\) representing combined growth factors over 3 and 4 hours.

Given: \(a=2\), \(m=3\), \(n=4\)

$$a^m \times a^n = 2^{3+4}$$

$$= 2^7$$

$$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$$

Answer: 128 growth factor

Example 2: Simplify \(\frac{5^6}{5^2}\) representing dilution factor after 6 and 2 successive dilutions.

Given: \(a=5\), \(m=6\), \(n=2\)

$$\frac{5^6}{5^2} = 5^{6-2} = 5^4$$

$$= 5 \times 5 \times 5 \times 5 = 625$$

Answer: 625 dilution factor

Example 3: Evaluate \((3^2)^4\) representing repeated enzyme activity increasing by power of 2, applied 4 times.

Given: \(a=3\), \(m=2\), \(n=4\)

$$(3^2)^4 = 3^{2 \times 4} = 3^8$$

$$= 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561$$

Answer: 6561 enzyme activity units

Example 4: Calculate \( (4^3)^2 \times 4^2 \) representing combined replication effects.

Given: \(a=4\), \(m=3\), \(n=2\), additional \(4^2\)

$$(4^3)^2 \times 4^2 = 4^{3 \times 2} \times 4^2 = 4^6 \times 4^2 = 4^{6+2} = 4^8$$

$$= 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 65536$$

Answer: 65536 replication factor

4.1.3 Indicial Equations

Indicial equations involve solving for unknown indices or bases in expressions involving powers. This skill is crucial in applied biology when determining growth rates or decay constants from experimental data.

The general approach is to express both sides with the same base and equate the indices.

Worked Examples

Example 1: Solve for \(x\) in the equation \(2^x = 32\), where 32 represents the population size.

Given: \(2^x = 32\)

Express 32 as a power of 2:

$$32 = 2^5$$

Equate indices:

$$x = 5$$

Answer: \(x = 5\)

Example 2: Find \(x\) if \(3^{2x} = 81\), representing enzyme concentration after \(2x\) hours.

Given: \(3^{2x} = 81\)

Express 81 as power of 3:

$$81 = 3^4$$

Equate indices:

$$2x = 4$$

$$x = \frac{4}{2} = 2$$

Answer: \(x = 2\) hours

Example 3: Solve \(5^{x+1} = 625\), modeling growth over \(x+1\) periods.

Given: \(5^{x+1} = 625\)

Express 625 as power of 5:

$$625 = 5^4$$

Equate indices:

$$x + 1 = 4$$

$$x = 4 - 1 = 3$$

Answer: \(x = 3\)

Example 4: Determine \(x\) if \(\frac{2^{x+2}}{2^{3}} = 16\), representing net growth factor.

Given: \(\frac{2^{x+2}}{2^{3}} = 16\)

Use quotient law:

$$2^{x+2 - 3} = 16$$

Simplify exponent:

$$2^{x - 1} = 16$$

Express 16 as power of 2:

$$16 = 2^4$$

Equate indices:

$$x - 1 = 4$$

$$x = 5$$

Answer: \(x = 5\)

Chapter Summary

This chapter covered the fundamental concepts of indices, starting with the definition of bases and progressing through the essential laws of indices that govern their manipulation. It then explored indicial equations, demonstrating how to solve equations involving powers. The study of logarithms followed, introducing the laws of logarithms which provide the rules for their calculation and simplification. Logarithmic operations were examined to show how to perform addition, subtraction, multiplication, and division using logarithms. The chapter also explained how to convert logarithms from one base to another, enabling flexibility in their application. Finally, it addressed the graphical representation of logarithmic and exponential functions, highlighting their behavior and key characteristics on coordinate axes. This comprehensive approach equips students with the tools to handle exponential and logarithmic expressions both algebraically and graphically.