Comprehensive Calculator Guide
A logarithm answers: to what power must the base be raised to give the number? log₁₀(1,000) = 3 because 10³ = 1,000, and log₂(1,024) = 10 because 2¹⁰ = 1,024. Enter the number and any positive base other than 1 (10, 2, e or your own) to see the result.
log, ln and changing base
log usually means base 10; ln is the natural log, base e ≈ 2.71828. Any log can be computed from another with the change-of-base rule, log_b(x) = ln(x) ÷ ln(b), which is how the calculator handles any base. For ln, enter 2.718281828 as the base: ln(20) = 2.9957.
The rules: log(ab) = log a + log b, log(a ÷ b) = log a − log b, log(aⁿ) = n log a, log_b(1) = 0 and log_b(b) = 1. Zero and negative numbers have no real logarithm, since a positive base to any power is positive.
Where logarithms show up
pH is minus the base-10 log of hydrogen-ion concentration, so 10⁻⁷ mol/L gives pH 7. Decibels and earthquake magnitudes are logarithmic scales. Doubling time at 7% growth is log base 1.07 of 2 = 10.24 years, close to the rule-of-72 estimate of 10.3. And log₂ tells you how many bits encode a set of options: 1,024 options need 10 bits.
How to Use
- 01Enter the number, which must be positive.
- 02Enter the base: 10, 2, 2.718281828 for ln, or any positive base other than 1.
- 03Read the power the base must be raised to.
Formula Used
log_b(x) = y ⟺ b^y = x | log_b(x) = ln(x) ÷ ln(b)Practical Examples
Example 1: common log
log₁₀(1,000) = 3.
Example 2: base 2
log₂(1,024) = 10: ten bits cover 1,024 options.
Example 3: doubling time
Number 2, base 1.07: 10.24 years to double at 7% a year.
Important Tips
- A negative result means the number is below 1: log₁₀(0.01) = −2.
- Check a result by raising the base to it with the exponent calculator.
Common Mistakes to Avoid
- Taking the log of zero or a negative number.
- Mixing up log (base 10) and ln (base e).
Frequently Asked Questions
- What is a logarithm, simply?
- The exponent that turns the base into the number: log₁₀(100) = 2 because 10² = 100.
- What is the difference between log and ln?
- log is usually base 10; ln is base e, about 2.71828.
- Why is there no log of a negative number?
- A positive base raised to any real power is positive, so no power gives zero or a negative.
- How long until money doubles?
- Take log base (1 + rate) of 2: at 7%, enter 2 and base 1.07 for 10.24 years.