Logarithm Calculator
Must be greater than 0
Common Log Values
ln(e)
1
log₁₀(10)
1
log₂(2)
1
log(1)
0
Log Types
Related Calculators
Understanding Logarithms
What is a Logarithm?
A logarithm answers the question: "What power must we raise the base to get a certain number?" If by = x, then logb(x) = y.
Example
23 = 8, so log₂(8) = 3
"2 raised to what power equals 8?" Answer: 3
Types of Logarithms
Natural Log (ln)
Base e (≈2.71828). Used extensively in calculus, physics, and natural sciences.
ln(e) = 1
Common Log (log₁₀)
Base 10. Used for pH scale, decibels, Richter scale, and general calculations.
log(10) = 1
Binary Log (log₂)
Base 2. Used in computer science for algorithm analysis and information theory.
log₂(2) = 1
Logarithm Laws
logb(xy) = logb(x) + logb(y) [Product Rule]
logb(x/y) = logb(x) - logb(y) [Quotient Rule]
logb(xn) = n × logb(x) [Power Rule]
logb(b) = 1 [Identity]
logb(1) = 0 [Zero Rule]
Change of Base Formula
To calculate a logarithm with any base using a calculator that only has ln or log:
logb(x) = ln(x) / ln(b) = log(x) / log(b)
Logarithms and Antilogarithms
Logarithms and exponentials are inverse functions:
Logarithm
Finding the exponent
log₁₀(100) = 2
Antilogarithm
Raising to the power
10² = 100
Real-World Applications
pH Scale
pH = -log₁₀[H⁺]
A logarithmic measure of acidity
Decibels
dB = 10 × log₁₀(P/P₀)
Logarithmic scale for sound intensity
Richter Scale
Each whole number = 10x more shaking
Magnitude 6 is 10x stronger than 5
Compound Interest
t = ln(A/P) / (n × ln(1 + r/n))
Time to reach investment goals
Important Notes
- • Logarithms are only defined for positive numbers (x > 0)
- • The base must be positive and not equal to 1
- • log(0) is undefined (approaches negative infinity)
- • log of a negative number is undefined in real numbers
Frequently Asked Questions
What is the difference between log and ln?
Log (common logarithm) uses base 10, while ln (natural logarithm) uses base e (approximately 2.71828). When you see 'log' without a base specified, it typically means log base 10. Natural log (ln) is used extensively in calculus, physics, and exponential growth/decay problems.
Why can't you take the logarithm of a negative number?
In real numbers, logarithms of negative numbers are undefined because no positive base raised to any real power can produce a negative result. For example, there's no real x where 10^x = -5. However, complex numbers do allow logarithms of negative numbers.
How do I calculate log with any base on a calculator?
Use the change of base formula: log_b(x) = ln(x)/ln(b) = log(x)/log(b). For example, to find log base 3 of 27: log₃(27) = log(27)/log(3) = 1.431/0.477 = 3. This works because 3³ = 27.
What are logarithms used for in real life?
Logarithms measure things that span huge ranges: earthquake magnitude (Richter scale), sound intensity (decibels), acidity (pH scale), and star brightness. They're also essential in finance for calculating compound interest growth time, and in computer science for algorithm complexity analysis.