LogBench

Logarithm Calculator

logₐ(n) = ln(n) / ln(a).

n (positive)

Base b

logₐ(n)

logₐ(n) = ln(n) / ln(a)

ln(n) — natural: 4.605
log₁₀(n): 2

What you entered

n (positive): 100Base b: 10

1
Natural log of n
ln(100)= 4.6052
2
Natural log of base 10
ln(10)= 2.3026
3
Change of base: divide
ln(100) ÷ ln(10)= 2

Result

logₐ(n): 2

log base 10 of 100 is 2.

Logarithms in common bases

A logarithm answers what power the base must be raised to in order to give the number. Natural log and log base 10 are shown for every row.

Expression	Result	Natural log	Log base 10
log10(1)	0.000000	0.000000	0.000000
log10(50)	1.698970	3.912023	1.698970
log10(100)	2.000000	4.605170	2.000000
log10(1000)	3.000000	6.907755	3.000000
log2(8)	3.000000	2.079442	0.903090
log2(1024)	10.000000	6.931472	3.010300

The log of 1 is always 0 in every base, since any base to the power 0 is 1. Logs convert multiplication into addition, which is what made slide rules and log tables work before calculators, and why log scales are used for anything spanning orders of magnitude - decibels, the Richter scale, pH. log2(1024) being exactly 10 is the same fact as 2^10 = 1024, read backwards. Logs of zero and negative numbers are undefined in the reals, which is a common source of errors when data contains zeros.

The inverse of exponentiation

logₐ(n) answers 'to what power must a be raised to get n?' — it undoes exponentiation the same way division undoes multiplication, and any base can be computed via the change-of-base formula using natural logs.

Where logs are essential

The Richter earthquake scale, decibels for sound, pH for acidity, and the time to double an investment (via the rule of 70) all rely on logarithms to compress huge ranges into manageable numbers.

Frequently asked questions

What is a logarithm?

logₐ(n) is the power you raise a to in order to get n. log₁₀(1000) = 3 because 10³ = 1000. ln(e) = 1 because e¹ = e. log₂(8) = 3 because 2³ = 8. It answers: 'how many times do I multiply a by itself to reach n?'

What is the change-of-base formula?

logₐ(n) = ln(n) / ln(a) = log₁₀(n) / log₁₀(a). This lets you compute any base logarithm using just natural log or log base 10. For example: log₅(100) = ln(100)/ln(5) = 4.605/1.609 = 2.861.

What are the key log rules?

Product: log(ab) = log(a) + log(b). Quotient: log(a/b) = log(a) − log(b). Power: log(aⁿ) = n×log(a). These rules convert multiplication into addition and exponents into multiplication — which is why slide rules worked.