Logarithms
📊 Master essential formulas with clear explanations, memory tricks, and practical examples. From basic arithmetic to advanced quantitative concepts, build a rock-solid foundation.
Important Formulas & Concepts
Study MaterialLogarithms
Logarithms is an important mathematical topic in Quantitative Aptitude and higher mathematics. Questions from logarithms are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, Defence, and various entrance examinations.
This chapter mainly deals with:
- Definition of logarithms
- Conversion between exponential and logarithmic forms
- Properties of logarithms
- Common logarithms
- Characteristic and mantissa
- Logarithmic equations and simplifications
Logarithms simplify complicated multiplication, division, powers, and roots into simple addition and subtraction operations.
What is a Logarithm?
If:
am = x
Then:
logax = m
where:
- a = base
- x = number
- m = logarithm value
Conditions for Logarithms
For logax to exist:
- a > 0
- a ≠1
- x > 0
Basic Examples of Logarithms
103 = 1000
Therefore:
log101000 = 3
25 = 32
Therefore:
log232 = 5
34 = 81
Therefore:
log381 = 4
Conversion Between Exponential and Logarithmic Forms
| Exponential Form | Logarithmic Form |
|---|---|
| am = x | logax = m |
| 23 = 8 | log28 = 3 |
| 102 = 100 | log10100 = 2 |
Important Properties of Logarithms
1. Product Rule
loga(xy) = logax + logay
Logarithm of multiplication becomes addition.
2. Quotient Rule
loga(x/y) = logax − logay
Logarithm of division becomes subtraction.
3. Power Rule
loga(xn) = n logax
4. Log of 1
loga1 = 0
because:
a0 = 1
5. Log of Base
logaa = 1
6. Reciprocal Property
logax = 1 / logxa
7. Change of Base Formula
logax = logbx / logba
or
logax = log x / log a
Important Formula Summary
| Property | Formula |
|---|---|
| Product Rule | loga(xy) = logax + logay |
| Quotient Rule | loga(x/y) = logax − logay |
| Power Rule | loga(xn) = nlogax |
| Log of 1 | loga1 = 0 |
| Log of Base | logaa = 1 |
| Reciprocal Rule | logax = 1/logxa |
| Change of Base | logax = log x/log a |
Common Logarithms
Logarithms with base 10 are called Common Logarithms.
Common Logarithm → Base 10
Example:
log 100 = log10100 = 2
Natural Logarithms
Logarithms with base e are called natural logarithms.
Natural Logarithm → Base e
Notation:
ln x
Characteristic and Mantissa
A logarithm has two parts:
- Characteristic
- Mantissa
1. Characteristic
The integer part of logarithm is called characteristic.
Case 1: Number Greater Than 1
Characteristic:
One less than number of digits before decimal point.
| Number | Characteristic |
|---|---|
| 654.2 | 2 |
| 26.5 | 1 |
| 8.23 | 0 |
Case 2: Number Less Than 1
Characteristic:
Negative and one more than zeros after decimal point.
| Number | Characteristic |
|---|---|
| 0.6 | -1 |
| 0.06 | -2 |
| 0.003 | -3 |
2. Mantissa
The decimal part of logarithm is called mantissa.
✔ Mantissa is always positive.
Important Logarithmic Values
| Expression | Value |
|---|---|
| log1010 | 1 |
| log10100 | 2 |
| log101000 | 3 |
| log28 | 3 |
| log381 | 4 |
Applications of Logarithms
- Simplification of calculations
- Exponential equations
- Scientific calculations
- Growth and decay models
- Compound interest problems
- Engineering and physics applications
Common Mistakes in Logarithms
- Using wrong logarithm properties.
- Ignoring logarithm conditions.
- Confusing base and power.
- Sign mistakes in characteristic.
- Calculation mistakes in change of base.
Important Exam Tips
- Memorize all logarithm properties.
- Practice conversion between exponential and logarithmic forms.
- Learn common logarithm values.
- Use product and quotient rules carefully.
- Understand characteristic and mantissa clearly.
- Simplify logarithmic expressions step-by-step.
- Verify bases carefully in calculations.
Quick Revision Table
| Concept | Formula |
|---|---|
| Definition | am = x ⇔ logax = m |
| Product Rule | log(xy) = log x + log y |
| Quotient Rule | log(x/y) = log x − log y |
| Power Rule | log(xn) = nlog x |
| log 1 | 0 |
| log a | 1 |
| Change of Base | logax = log x/log a |
Logarithms is an important chapter for simplifying complex calculations and solving advanced mathematical problems. Strong understanding of logarithmic properties and formulas helps candidates solve questions quickly and accurately in competitive examinations.