SETS

Laws of Indices
1. Multiplication Law
When multiplying powers with the same base, add the indices.

2. Division Law
When dividing powers with the same base, subtract the indices.

3. Power of a Power
Multiply the indices.

4. Zero Index
Any non-zero number raised to power zero equals 1.

5. Negative Index
A negative index means take the reciprocal.

6. Fractional Indices
Therefore:
and
B. Applications of Indices
Indices are useful when dealing with:
Very large or very small numbers
Scientific notation
Population growth
Compound interest
Computer storage and data
Area and volume calculations
Example
Write 5,000,000 in standard form:
C. Exponential Equations
An exponential equation has the unknown in the exponent.
Example 1
Since:
Therefore:
So:
Exponential Equations in Quadratic Form
Consider:
Notice that:
Let:
Then:
Factorise:
Therefore:
Since �:
or
Therefore:
Teaching tip: Emphasise that the substitution � changes the exponential equation into an ordinary quadratic equation.
D. Introduction to Logarithms
Explain to students:
A logarithm tells us the power to which a number must be raised to obtain another number.
For example:
can be written as:
Read this as:
"The logarithm of 8 to base 2 is 3."
General relationship
E. Laws of Logarithms
1. Product Law
Example:
2. Quotient Law
3. Power Law
Example:
F. Change of Base
The change-of-base formula is:
A common choice is base 10:
Example
Evaluate:
Using change of base:
Using a calculator: