Logarithm Calculator
Calculate log base 10, natural log, and any custom base logarithm.
The log argument must be greater than zero.
Logarithm result
log(1000) = 3
log(x) = log₁₀(x)
Argument x
1,000
Base
10
Common log
Exponential equivalent
10^3 = 1,000
Inverse check
1,000
base^result should return x.
Change of base
Not needed
Common and natural logs are direct.
Precision
6 decimals
Logarithm Formulas and Rules
Logarithm definition
log_b(x) = y means b^y = x
Common log
log(x) = log_10(x)
Natural log
ln(x) = log_e(x)
Change of base
log_b(x) = ln(x) ÷ ln(b)
Change of base with log
log_b(x) = log(x) ÷ log(b)
Product rule
log_b(MN) = log_b(M) + log_b(N)
Quotient rule
log_b(M / N) = log_b(M) − log_b(N)
Power rule
log_b(M^k) = k × log_b(M)
Variable Explanations
b
Logarithm base.
x
Argument or input value.
y
Logarithm result, which is an exponent.
e
Euler's number, approximately 2.71828.
log(x)
Commonly means base 10 in calculators.
ln(x)
Means base e.
Base rule
Base must be positive and not equal to 1.
Argument rule
Argument must be positive.
What Logarithms Mean
Inverse of exponents
Logarithms undo exponentiation.
Power question
log_b(x) asks: what power of b gives x?
Solving exponents
Logs help solve equations where the unknown is in the exponent.
Large ranges
Logs can compress large ranges of numbers.
Useful fields
Logs appear in science, finance, computer science, and data scales.
Example
log2(32) asks: 2 to what power equals 32? Answer: 5.
Worked Examples
log10(100)
Rule: Common log
Equivalent: 10² = 100
Answer: 2
Base 10 raised to 2 equals 100.
log10(1000)
Rule: Common log
Equivalent: 10³ = 1000
Answer: 3
Powers of 10 are easy common-log examples.
ln(e)
Rule: Natural log
Equivalent: e¹ = e
Answer: 1
Natural log uses base e.
log2(8)
Rule: Custom base
Equivalent: 2³ = 8
Answer: 3
2 multiplied by itself 3 times equals 8.
log3(81)
Rule: Custom base
Equivalent: 3⁴ = 81
Answer: 4
The result is the exponent needed on base 3.
log5(1)
Rule: Zero exponent rule
Equivalent: 5⁰ = 1
Answer: 0
Any valid base raised to 0 equals 1.
log7(20)
Rule: Change of base
Equivalent: ln(20) ÷ ln(7)
Answer: ≈ 1.539
Change of base handles non-obvious values.
Solve 2^x = 32
Rule: Log inverse
Equivalent: x = log2(32)
Answer: 5
Logs solve equations where the unknown is an exponent.
Common Logarithm vs Natural Logarithm
Common logarithm
Common log uses base 10. It appears often in powers of 10, pH, decibels, and scientific notation contexts.
Natural logarithm
Natural log uses base e. It appears often in exponential growth, decay, and continuous compounding.
Logarithm Rules and Change of Base
Product rule
Turns multiplication into addition.
Quotient rule
Turns division into subtraction.
Power rule
Moves an exponent to the front.
Change of base
Lets you calculate any base using ln or log.
Common Logarithm Mistakes
Understanding Your Result
Common log result
The exponent needed for base 10.
Natural log result
The exponent needed for base e.
Custom log result
The exponent needed for the chosen base.
Exponential equivalent
Confirms the log answer by reversing it.
Change-of-base form
Shows how custom-base logs are computed.