Logarithm Calculator
Calculate logarithms with any base (log₁₀, ln, or custom base). Shows step-by-step solution, the inverse exponential form, and logarithm properties with examples.
Logarithm Calculator
log₍10.00₎(100) = ?
log base 10.00 of 100
2
Logarithm Table (Base 10.00)
| Value | Logarithm |
|---|---|
| 0.1 | -1 |
| 0.5 | -0.30103 |
| 1 | 0 |
| 2 | 0.30103 |
| 5 | 0.69897 |
| 10 | 1 |
| 50 | 1.69897 |
| 100 | 2 |
| 1000 | 3 |
Logarithm Rules
- log_b(xy) = log_b(x) + log_b(y)
- log_b(x/y) = log_b(x) - log_b(y)
- log_b(x^n) = n * log_b(x)
- log_b(1) = 0
- log_b(b) = 1
- log_a(x) = log_b(x) / log_b(a)
What are Logarithms?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise the base to get this number?"
Definition: If b^x = y, then log_b(y) = x
In Words: "The logarithm base b of y is x" means "b raised to the power x equals y"
Common Types of Logarithms:
- Common Logarithm (log or log₁₀): Base 10 - used in pH, decibels, earthquake scales
- Natural Logarithm (ln or logₑ): Base e ≈ 2.71828 - used in calculus, growth/decay
- Binary Logarithm (log₂): Base 2 - used in computer science, information theory
Key Values to Remember:
- log_b(1) = 0 for any base b (because b⁰ = 1)
- log_b(b) = 1 for any base b (because b¹ = b)
- log_b(bⁿ) = n (logs and exponents are inverses)
- Logarithms of negative numbers are undefined in real numbers
Logarithm Properties and Rules
These properties are essential for simplifying logarithmic expressions:
Fundamental Logarithm Properties
Where:
- b= Base of the logarithm (b > 0, b ≠ 1)
- x, y= Positive real numbers
- n= Any real number
Common Logarithm Values
Reference table for frequently used logarithm values:
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.693 | 1 |
| e ≈ 2.718 | 0.434 | 1 | 1.443 |
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| 1000 | 3 | 6.908 | 9.966 |
Useful Relationships:
- ln(x) ≈ 2.303 × log₁₀(x)
- log₁₀(x) ≈ 0.434 × ln(x)
- log₂(x) ≈ 3.322 × log₁₀(x)
Change of Base Formula
The change of base formula allows you to calculate logarithms with any base using a calculator that only has log or ln:
Change of Base Formula
Where:
- b= Original base you want
- c= Available base (usually 10 or e)
- x= The argument of the logarithm
How to Use This Logarithm Calculator
Our calculator computes logarithms with any base:
- Enter the Number: The value you want to find the logarithm of (must be positive)
- Select or Enter Base:
- Common log (base 10): Select "log" or enter 10
- Natural log (base e): Select "ln" or enter e
- Binary log (base 2): Enter 2
- Any custom base: Enter your desired base
- View Results: Get the logarithm value and step-by-step solution
Additional Features:
- Expand logarithmic expressions using properties
- Condense multiple logarithms into one
- Solve logarithmic equations
- Convert between different bases
Solving Logarithmic Equations
Strategies for solving equations involving logarithms:
Method 1: Convert to Exponential Form
- If log_b(x) = c, then x = b^c
- Example: log₃(x) = 4 → x = 3⁴ = 81
Method 2: Use Logarithm Properties
- Combine or expand logs, then solve
- Example: log(x) + log(x-3) = 1
- log(x(x-3)) = 1 → x(x-3) = 10 → x² - 3x - 10 = 0
Method 3: One-to-One Property
- If log_b(x) = log_b(y), then x = y
- Example: log₂(x+1) = log₂(5) → x+1 = 5 → x = 4
Important: Always check your solutions! Logarithms are only defined for positive arguments, so verify that your answer doesn't make any log argument negative or zero.
Real-World Applications of Logarithms
Logarithms appear throughout science and everyday life:
Sound (Decibels):
- dB = 10 × log₁₀(P/P₀)
- Doubling sound power adds 3 dB
- 10× power adds 10 dB
Chemistry (pH Scale):
- pH = -log₁₀[H⁺]
- Each pH unit = 10× change in acidity
- pH 3 is 10× more acidic than pH 4
Earthquakes (Richter Scale):
- Each whole number = 10× amplitude increase
- Magnitude 6 is 10× stronger than magnitude 5
Finance (Compound Interest):
- Time to double money: t = ln(2)/r ≈ 0.693/r
- Rule of 72: Years to double ≈ 72/interest rate
Computer Science:
- Binary search: O(log₂ n) operations
- Information content measured in bits (log₂)
- Algorithm complexity analysis
Worked Examples
Calculate Logarithm Using Definition
Problem:
Find log₂(32)
Solution Steps:
- 1Ask: 2 to what power equals 32?
- 2Test powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32
- 32⁵ = 32
- 4Therefore, log₂(32) = 5
Result:
log₂(32) = 5
Use Change of Base Formula
Problem:
Calculate log₅(125) using common logarithms
Solution Steps:
- 1Apply change of base: log₅(125) = log(125) / log(5)
- 2Calculate log(125) = 2.097
- 3Calculate log(5) = 0.699
- 4Divide: 2.097 / 0.699 = 3
- 5Verify: 5³ = 125 ✓
Result:
log₅(125) = 3
Expand Logarithmic Expression
Problem:
Expand log₃(xy²/z)
Solution Steps:
- 1Apply quotient rule: log₃(xy²) - log₃(z)
- 2Apply product rule to first term: log₃(x) + log₃(y²) - log₃(z)
- 3Apply power rule: log₃(x) + 2log₃(y) - log₃(z)
- 4Final expanded form
Result:
log₃(xy²/z) = log₃(x) + 2log₃(y) - log₃(z)
Tips & Best Practices
- ✓log and 10^x are inverses; ln and e^x are inverses
- ✓Use change of base formula to calculate any logarithm on a basic calculator
- ✓When expanding, logs of products → sums, logs of quotients → differences
- ✓When condensing, sums → logs of products, differences → logs of quotients
- ✓Always check solutions in logarithmic equations—arguments must be positive
- ✓log(10^n) = n and 10^(log x) = x — these inverse relationships are key
- ✓For rough estimates: log(2) ≈ 0.3, log(3) ≈ 0.5, ln(2) ≈ 0.7
Frequently Asked Questions
Sources & References
Last updated: 2026-01-22
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: Handbook of Mathematical Functions
by Abramowitz & Stegun