Indices and Logarithms

1. Laws of Indices

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

2. Laws of Logarithms

Laws of Logarithms

Logarithms are the inverse operations of exponents. If:

$$\log_b x = y \quad \text{then} \quad b^y = x$$logb​x=ythenby=x

where $b$b is the base, $x$x is the number, and $y$y is the logarithm.

The fundamental laws of logarithms are as follows:

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1. Product Law

$$\log_b (MN) = \log_b M + \log_b N$$logb​(MN)=logb​M+logb​N

Example:

$$\log_2 (8 \times 4) = \log_2 8 + \log_2 4$$log2​(8×4)=log2​8+log2​4

Since $\log_2 8 = 3$log2​8=3 and $\log_2 4 = 2$log2​4=2, we get:

$$\log_2 32 = 3 + 2 = 5$$log2​32=3+2=5

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2. Quotient Law

$$\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$$logb​(NM​)=logb​M−logb​N

Example:

$$\log_3 \left( \frac{81}{9} \right) = \log_3 81 - \log_3 9$$log3​(981​)=log3​81−log3​9

Since $\log_3 81 = 4$log3​81=4 and $\log_3 9 = 2$log3​9=2, we get:

$$\log_3 9 = 4 - 2 = 2$$log3​9=4−2=2

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3. Power Law

$$\log_b (M^n) = n \log_b M$$logb​(Mn)=nlogb​M

Example:

$$\log_5 (25^3) = 3 \log_5 25$$log5​(253)=3log5​25

Since $\log_5 25 = 2$log5​25=2, we get:

$$\log_5 (25^3) = 3 \times 2 = 6$$log5​(253)=3×2=6

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4. Change of Base Law

$$\log_b M = \frac{\log_k M}{\log_k b}$$logb​M=logk​blogk​M​

This is useful for converting logarithms into a different base, commonly base 10 (logarithm) or base $e$e (natural logarithm, denoted as $\ln$ln).

Example: Convert $\log_2 10$log2​10 to base 10.

$$\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2}$$log2​10=log10​2log10​10​

Since $\log_{10} 10 = 1$log10​10=1 and $\log_{10} 2 \approx 0.301$log10​2≈0.301, we get:

$$\log_2 10 = \frac{1}{0.301} \approx 3.32$$log2​10=0.3011​≈3.32

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5. Logarithm of 1

$$\log_b 1 = 0$$logb​1=0

Because any number raised to the power of zero equals 1:

$$b^0 = 1$$b0=1

Example:

$$\log_7 1 = 0$$log7​1=0

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6. Logarithm of the Base

$$\log_b b = 1$$logb​b=1

Because $b^1 = b$b1=b.
Example:

$$\log_4 4 = 1$$log4​4=1

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7. Natural Logarithms (Base $e$e)

$$\ln x = \log_e x$$lnx=loge​x

Natural logarithms use the base $e$e (Euler’s number, approximately 2.718). The same logarithmic laws apply:

$$\ln (MN) = \ln M + \ln N$$ln(MN)=lnM+lnN $$\ln \left( \frac{M}{N} \right) = \ln M - \ln N$$ln(NM​)=lnM−lnN $$\ln (M^n) = n \ln M$$ln(Mn)=nlnM

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Examples Using Multiple Laws

1. Simplify $\log_3 81 + \log_3 9$log3​81+log3​9

   • Using the product law:

     $$\log_3 (81 \times 9) = \log_3 729$$log3​(81×9)=log3​729

   • Since $3^6 = 729$36=729, we get:

     $$\log_3 729 = 6$$log3​729=6

2. Simplify $\log_5 125 - \log_5 25$log5​125−log5​25

   • Using the quotient law:

     $$\log_5 \left( \frac{125}{25} \right) = \log_5 5$$log5​(25125​)=log5​5

   • Since $\log_5 5 = 1$log5​5=1, the answer is:

     $$1$$1

3. Simplify $2 \log_4 8$2log4​8

   • Using the power law:

     $$\log_4 8^2 = \log_4 64$$log4​82=log4​64

   • Since $4^3 = 64$43=64, we get:

     $$\log_4 64 = 3$$log4​64=3

3. Tutor-Marked Assignment

Section A: Indices (Laws of Exponents)

1. Simplify the following expressions:
   a) $3^4 \times 3^2$34×32
   b) $\frac{5^7}{5^4}$5457​
   c) $(2^3)^4$(23)4
   d) $8^{2/3}$82/3

2. Solve for $x$x in the following equations:
   a) $2^x = 32$2x=32
   b) $5^{x+1} = 125$5x+1=125
   c) $9^{x-1} = 27^{x+2}$9x−1=27x+2

3. Express in simplest form:
   a) $(a^3b^{-2})^4$(a3b−2)4
   b) $\frac{x^5 y^{-3}}{x^{-2} y^4}$x−2y4x5y−3​

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Section B: Logarithms (Laws of Logarithms)

4. Evaluate:
   a) $\log_2 32$log2​32
   b) $\log_{10} 1000$log10​1000
   c) $\log_5 25 + \log_5 5$log5​25+log5​5

5. Solve for $x$x:
   a) $\log_3 x = 4$log3​x=4
   b) $\log_2 (x + 1) = 3$log2​(x+1)=3
   c) $\log_5 (x^2 - 4) = \log_5 5$log5​(x2−4)=log5​5

6. Express as a single logarithm:
   a) $\log_a x + \log_a y$loga​x+loga​y
   b) $2\log_b m - \frac{1}{2} \log_b n$2logb​m−21​logb​n

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Bonus Question:

Prove that:
$\log_a (mn) = \log_a m + \log_a n$loga​(mn)=loga​m+loga​n
using the laws of logarithms.