What is a Exponent & Logarithm Calculator?
This calculator computes exponents and logarithm values.
How it's calculated
aĖ£ = b ā log_a b = x
Common logarithm logāā, natural logarithm ln (base e ā 2.71828)
Step-by-step example
Evaluating logā 32
- Step 1 Reframe as "2 raised to what power equals 32?"
- Step 2 Confirm 2āµ = 32
- Step 3 Result: logā 32 = 5
Interpretation: A logarithm is the inverse of exponentiation ā it finds the power needed to reach a given value.
Logarithms turn multiplication into addition
log(AB) = log A + log B ā before calculators existed, this was revolutionary. Astronomers needing to multiply 12-digit numbers could use logarithm tables to turn multiplication into simple addition, saving enormous amounts of time. The mathematician Laplace reportedly said logarithms 'doubled the life of an astronomer' by cutting the time spent on calculation.
Good to know
- Many of our senses are logarithmic. Sound level (dB), earthquake magnitude (Richter scale), and pH are all logarithmic ā a Richter magnitude 7 releases about 32 times more energy than magnitude 6, not 10 times.
- The natural constant e emerged from compound interest ā if you compound 100% annual interest infinitely often, the result converges to e ā 2.71828.
- eĖ£ is its own derivative ā the only function where differentiating gives back the exact same function, which is why it holds a special place in calculus.
- Euler's identity, e^(iĻ) + 1 = 0, is often cited as the most beautiful equation in mathematics.
- Slide rules, worn around the necks of engineers well into the mid-20th century, worked entirely on logarithmic scales.
Frequently asked questions
Q. What's the difference between log and ln?
log usually means base 10; ln is the natural logarithm with base e.
Q. What if the argument of a log is negative?
Undefined ā no real exponent of a positive base produces a negative result.
Q. How do I change the base of a logarithm?
Use the change-of-base formula: log_a b = log_c b Ć· log_c a.