Logarithm Formulas Sheet | Properties of Logarithms | Log Rules

In this session explained about logarithm formulas ( Laws of Logarithms) like logarithm addition rule and subtract rule and Base change rules and results on logarithmic inequalities with examples

Table of Contents

Log Formula | Logarithm Rules Practice | Logarithm Tutorial | Exercise – 2

Please go through the below link for basic concepts of logarithms like Definition of Logarithm , Characteristic and Mantissa , Rule for write Mantissa and Characteristic , Natural Logarithm , Common Logarithms .. etc

Logarithm Tutorial | Exercise – 1

Logarithm formula sheet ( Laws of Logarithms )

First Law of the logarithms – ( logarithm addition rule )

” Log of product = Sum of Logs”

log _am n = log _am + log _an ;

Where m, n & a are positive real numbers and a ≠ 1

For example – Take 524 x 254

Let 524 = 10ⁿ and 254 = 10^m

Now rewrite above exponential forms into logarithmic forms

i.e n = log ₁₀ 524 and m = log ₁₀ 254

⇒ 524 x 254 = 10ⁿx 10^m

⇒ 524 x 254 = 10 ^n+m

Now rewrite the above exponential forms into logarithmic forms

n + m = log ₁₀ (524 x 254)

Substitute ‘n‘ and ‘m‘ vales in the above equation

log ₁₀ 524 + log ₁₀ 254 = log ₁₀ (524 x 254)

Generalization of the above logarithmic law

log (abc) = log a + log b + log c

log (n₁n₂n₃n₄n₅ – – – – – – – n_k) = log n₁ + log n₂ + log n₃ + log n₄ + log n₅ + – – – – – – + log n_k

(Note: As per log rules, If the base is not mentioned for any logarithmic equations, then it is considered as base 10. The logarithmic calculation to base 10 is called common logarithms.)

Second Law of the logarithms – ( logarithm subtract rule )

” log of quotient = difference of logs”

log _a(m/n) = log _am – log _an ;

Where m, n & a are positive real numbers and a ≠ 1

For example – Take 524 / 254

Let 524 = 10^m and 254 = 10ⁿ

Now rewrite the above exponential forms into logarithmic forms

i.e m = log ₁₀ 524 and n = log ₁₀ 254

⇒ 524 /254 = 10^mx 10ⁿ

⇒ 524 / 254 = 10 ^{m -n}

Now rewrite the above exponential forms into logarithmic forms

m – n = log ₁₀ (524 /254)

Substitute ‘n‘ and ‘m‘ values in the above equation

log ₁₀ 524 – log ₁₀ 254 = log ₁₀ (524/254)

Third Law of the logarithms –

log _a(m ⁿ) = n log _am ;

Where m, n, and a are positive real numbers and a ≠ 1

For example – Take 52⁸

Let 52 = 10ⁿ —— ( 1)

Now rewrite the above exponential forms into logarithmic forms

i.e n = log ₁₀ 52 —– ( 2)

Here 52 can be written as 5.2 x 10¹

Now 52⁸ = (5.2 x 10¹ )⁸ = 5.2⁸ x 10⁸

Raising the power to 8 on both sides of the equation ( 1)

52⁸ = (10ⁿ)⁸

⇒ 52⁸ = 10⁸ⁿ

Now rewrite the above exponential forms into logarithmic forms

8 n = log ₁₀ (52⁸)

Substituting the value of ‘n’ in the above equation

8 log ₁₀ 52 = log ₁₀ (52⁸)

Base Change Rules

1. $\log_{n} m = \frac{1}{\log_{m} n}$

2. log _n m = log _a m / log _a n = log _a m . log _n a

3. $\log_{(n^b)} m = \frac{1}{b} \log_{n} m$

Some other Properties of logarithms

3. a log _a N = N

4. log _a a = 1

5. $\log_{n} m = \frac{\log_{10} m}{\log_{10} n} = \frac{\log m}{\log n}$

6. $\log_{(n^b)}(n^a) = \frac{a}{b}$

7. $\log_{(n^b)}(m^a) = a \log_{(n^b)}(m) = \frac{a}{b} \log_{n}(m)$

Logarithmic Inequalities

1. If n > 1 and log _n a > log _n b then a > b

2. If n <1 1 and log _n a > log _n b then a < b

Please go through the below link for logarithm applications with examples and solutions

Logarithm Tutorial | Exercise – 3

Logarithm formulas sheet| log rules |properties of logarithms | logarithm rules practice | logarithm tutorial

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