Logarithms
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Solved Examples
Study MaterialSolved Examples – Logarithms
Solved examples help students understand the practical application of logarithm concepts in competitive examinations. These examples are designed from basic to advanced level and cover important questions frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, Defence, and various entrance examinations.
Topics Covered in Solved Examples
- Basic logarithm evaluation
- Conversion between exponential and logarithmic forms
- Product and quotient rules
- Power rule applications
- Change of base formula
- Characteristic and mantissa
- Logarithmic equations
- Advanced logarithmic simplifications
Example 1: Basic Logarithm Evaluation
Question: Evaluate log232.
Solution:
Let:
log232 = x
Then:
2x = 32
Since:
32 = 25
Therefore:
x = 5
Answer: 5
Example 2: Converting Exponential Form
Question: Convert 103 = 1000 into logarithmic form.
Solution:
Using:
am = x ⇔ logax = m
Therefore:
log101000 = 3
Answer: log101000 = 3
Example 3: Product Rule Application
Question: Simplify log 5 + log 2.
Solution:
Using product rule:
log x + log y = log(xy)
Therefore:
log 5 + log 2
= log(5 × 2)
= log 10
= 1
Answer: 1
Example 4: Quotient Rule Application
Question: Simplify log 100 − log 10.
Solution:
Using quotient rule:
log x − log y = log(x/y)
Therefore:
log 100 − log 10
= log(100/10)
= log 10
= 1
Answer: 1
Example 5: Power Rule Application
Question: Simplify log(24).
Solution:
Using power rule:
log(xn) = nlog x
Therefore:
log(24)
= 4log 2
Answer: 4log 2
Example 6: Evaluating Common Logarithm
Question: Find log1010000.
Solution:
10000 = 104
Therefore:
log1010000 = 4
Answer: 4
Example 7: Change of Base Formula
Question: Express log28 using common logarithm.
Solution:
Using change of base formula:
logax = log x / log a
Therefore:
log28 = log 8 / log 2
Answer: log 8 / log 2
Example 8: Reciprocal Property
Question: If log35 = x, find log53.
Solution:
Using reciprocal property:
logax = 1/logxa
Therefore:
log53 = 1/x
Answer: 1/x
Example 9: Solving Logarithmic Equation
Question: Solve log2x = 4.
Solution:
Convert into exponential form:
x = 24
= 16
Answer: x = 16
Example 10: Simplifying Combined Logarithms
Question: Simplify log 2 + log 5 + log 10.
Solution:
Using product rule:
= log(2 × 5 × 10)
= log 100
= 2
Answer: 2
Example 11: Finding Characteristic
Question: Find the characteristic of log 542.8.
Solution:
Digits before decimal:
= 3
Characteristic:
= 3 − 1
= 2
Answer: 2
Example 12: Characteristic for Decimal Number
Question: Find the characteristic of log 0.0045.
Solution:
Zeros after decimal before first digit:
= 2
Characteristic:
= -(2 + 1)
= -3
Answer: -3
Example 13: Evaluating Nested Logarithm
Question: Evaluate log2(log381).
Solution:
First:
log381 = 4
Now:
log24 = 2
Answer: 2
Example 14: Simplifying Logarithmic Fraction
Question: Simplify log(1/100).
Solution:
1/100 = 10-2
Therefore:
log(1/100)
= log(10-2)
= -2
Answer: -2
Example 15: Advanced Simplification
Question: Simplify:
2log 3 + log 5 − log 9
Solution:
Using power rule:
2log 3 = log 32
= log 9
Expression:
= log 9 + log 5 − log 9
= log 5
Answer: log 5
Example 16: Solving Advanced Equation
Question: Solve log5(x − 1) = 2.
Solution:
Convert into exponential form:
x − 1 = 52
x − 1 = 25
x = 26
Answer: x = 26
Example 17: Using Change of Base
Question: Evaluate:
(log 64)/(log 2)
Solution:
Using change of base:
(log 64)/(log 2)
= log264
= 6
Answer: 6
Example 18: Evaluating Logarithmic Identity
Question: Find the value of:
log71
Solution:
Using identity:
loga1 = 0
Therefore:
log71 = 0
Answer: 0
Important Exam Tips
- Memorize all logarithmic properties.
- Practice converting exponential and logarithmic forms.
- Learn important powers and roots.
- Use product and quotient rules carefully.
- Understand characteristic and mantissa properly.
- Simplify expressions step-by-step.
- Verify logarithm bases carefully.
Practicing solved examples regularly improves conceptual clarity, logical thinking, and calculation speed in solving logarithm questions in competitive examinations.