Logarithm Calculator

Calculate logarithms and antilogarithms with any base. Find log, ln, and log base 2.

Logarithm Calculator

Mode

Number (x)

Base (b)

Common Bases

log₍10.00₎(100) = ?

log base 10.00 of 100

loglog₁₀

lnln (natural)

4.60517

lblog₂

6.643856

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

Product Rule: log_b(xy) = log_b(x) + log_b(y) Quotient Rule: log_b(x/y) = log_b(x) - log_b(y) Power Rule: log_b(x^n) = n × log_b(x) Change of Base: log_b(x) = log_c(x) / log_c(b) = ln(x) / ln(b) = log(x) / log(b) Inverse Properties: b^(log_b(x)) = x log_b(b^x) = x

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

log_b(x) = log_c(x) / log_c(b) Using common logarithm (base 10): log_b(x) = log(x) / log(b) Using natural logarithm (base e): log_b(x) = ln(x) / ln(b) Example: log₅(25) = log(25) / log(5) = 1.398 / 0.699 = 2

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

log (common logarithm) uses base 10, while ln (natural logarithm) uses base e ≈ 2.71828. Use log for practical scales (decibels, pH) and base-10 calculations. Use ln for calculus, continuous growth/decay, and theoretical mathematics. They're related: ln(x) = log(x) × 2.303.

In real numbers, logarithms of negative numbers are undefined because no real power of a positive base gives a negative result. For example, there's no real x where 10^x = -5. However, in complex analysis, logarithms of negative numbers do exist and involve imaginary numbers.

Convert to exponential form: if log(x) = 3, then x = 10³ = 1000. In general, log_b(x) = c means x = b^c. Always verify by substituting back: log(1000) = log(10³) = 3 ✓

The natural logarithm (ln) is fundamental in calculus because d/dx[ln(x)] = 1/x - the simplest non-trivial derivative. It's used in exponential growth/decay (population, radioactive decay), compound interest with continuous compounding, probability distributions, and throughout physics and engineering.

On a logarithmic scale, equal distances represent equal ratios rather than equal differences. Moving from 1 to 10 is the same distance as 10 to 100, because both represent multiplication by 10. This compresses large ranges and emphasizes relative changes—ideal for data spanning many orders of magnitude.

log₂ (binary logarithm) counts how many times you can halve a number before reaching 1—essentially counting binary digits. It appears in algorithm analysis (binary search is O(log n)), measuring information in bits, and calculating tree heights. log₂(1024) = 10, meaning 1024 = 2¹⁰ (1 KB).

Sources & References

Wolfram MathWorld - Logarithm (2024)
Khan Academy - Logarithms (2024)
MIT OpenCourseWare - Single Variable Calculus (2024)
Purplemath - Logarithms (2024)

Last updated: 2026-01-22

Logarithm Calculator