Indices and Logarithms

Laws of Indices (Exponents)

Indices (or exponents) are used to express repeated multiplication of a number. If $a$a is a real number and $m, n$m,n are integers, then the laws of indices are as follows:

1. Product Law

$$a^m \times a^n = a^{m+n}$$am×an=am+n

Example:

$$2^3 \times 2^4 = 2^{3+4} = 2^7$$23×24=23+4=27

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2. Quotient Law

$$\frac{a^m}{a^n} = a^{m-n}, \quad \text{where } a \neq 0$$anam​=am−n,where a=0

Example:

$$\frac{5^6}{5^2} = 5^{6-2} = 5^4$$5256​=56−2=54

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3. Power of a Power Law

$$(a^m)^n = a^{m \times n}$$(am)n=am×n

Example:

$$(3^2)^4 = 3^{2 \times 4} = 3^8$$(32)4=32×4=38

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4. Power of a Product Law

$$(ab)^m = a^m \times b^m$$(ab)m=am×bm

Example:

$$(2 \times 3)^4 = 2^4 \times 3^4$$(2×3)4=24×34

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5. Power of a Quotient Law

$$\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}, \quad \text{where } b \neq 0$$(ba​)m=bmam​,where b=0

Example:

$$\left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3} = \frac{64}{125}$$(54​)3=5343​=12564​

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6. Zero Index Law

$$a^0 = 1, \quad \text{where } a \neq 0$$a0=1,where a=0

Example:

$$7^0 = 1$$70=1

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7. Negative Index Law

$$a^{-m} = \frac{1}{a^m}, \quad \text{where } a \neq 0$$a−m=am1​,where a=0

Example:

$$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$3−2=321​=91​

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8. Fractional Indices (Roots Law)

$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$anm​=nam​=(na​)m

Example:

$$16^{\frac{1}{2}} = \sqrt{16} = 4$$1621​=16​=4 $$27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$$2732​=(327​)2=32=9

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Examples Using Multiple Laws

1. Simplify $2^3 \times 2^{-1}$23×2−1

   • Using the product law:

     $$2^{3+(-1)} = 2^2 = 4$$23+(−1)=22=4

2. Simplify $\frac{5^4}{5^6}$5654​

   • Using the quotient law:

     $$5^{4-6} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$54−6=5−2=521​=251​

3. Simplify $(4^2)^3$(42)3

   • Using the power of a power law:

     $$4^{2 \times 3} = 4^6 = 4096$$42×3=46=4096