Logarithm Calculator

A logarithm is the inverse of exponentiation. If b^y = x, then log_b(x) = y. The question a logarithm asks is: "What power do I raise b to in order to get x?" For example, log₁₀(1000) = 3 because 10³ = 1000. That single relationship — exponent and logarithm as mirror operations — underpins almost every logarithm calculation you will ever do.

Logarithms appear in an enormous range of fields. The Richter scale measures earthquake energy on a base-10 log scale. Sound intensity in decibels uses log₁₀. Chemical pH — the measure of acidity — is the negative log₁₀ of hydrogen ion concentration. The natural log (base e) governs continuous growth and decay in biology, finance, and physics. And the binary log (base 2) is the backbone of computer science, from algorithm complexity to data compression to information entropy.

Despite this breadth, there are really only a handful of formulas to know: the definition, the change-of-base formula, and the three arithmetic rules — product, quotient, and power. This calculator works through all of them for any input you provide.

When scientists and engineers write "log" without a base, they almost always mean log base 10 — the common logarithm. It is the natural choice for scales that span many orders of magnitude, because each unit increase represents a tenfold change.

pH chemistry: pH = −log₁₀([H⁺]). A solution with [H⁺] = 0.001 mol/L has pH = −log₁₀(0.001) = 3. Each integer step on the pH scale is a 10× change in hydrogen ion concentration. Sound (decibels): L = 10 × log₁₀(I/I₀), where I₀ is the threshold of hearing. A 10 dB increase means 10× more intensity. Earthquakes (Richter): each unit increase represents roughly 31.6× more energy released.

Key values to memorize: log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3, log₁₀(0.1) = −1. These follow directly from the definition: 10^n = 10ⁿ.

The natural logarithm uses base e ≈ 2.71828, Euler's number. It is "natural" because it arises spontaneously in calculus: the derivative of ln(x) is simply 1/x, which makes it the ideal tool for integration, differential equations, and growth models.

Continuous compounding: if you invest $1 at 100% annual interest compounded continuously, after one year you have e ≈ $2.718. The formula A = Pe^(rt) uses ln to solve for time: t = ln(A/P) / r. Population growth: N(t) = N₀·e^(rt) — the natural log extracts the growth rate or time from any exponential process. Entropy: information entropy in physics is defined using ln.

Key values: ln(1) = 0, ln(e) = 1, ln(e²) = 2, ln(e^n) = n. The natural log is the inverse of the exponential function: ln(e^x) = x and e^(ln x) = x.

The binary logarithm counts the number of times you can halve a quantity before reaching 1, or equivalently, the number of bits needed to represent n distinct values. It is fundamental to computer science, information theory, and algorithm analysis.

Algorithm complexity: binary search halves the search space each step, so a search over n items takes log₂(n) comparisons. Sorting n items takes O(n log₂ n) comparisons. Information entropy: a fair coin toss carries log₂(2) = 1 bit of information; choosing uniformly from 8 options carries log₂(8) = 3 bits. Data structures: a balanced binary tree of n nodes has depth ≈ log₂(n).

Key values: log₂(1) = 0, log₂(2) = 1, log₂(4) = 2, log₂(8) = 3, log₂(16) = 4, log₂(1024) = 10. Powers of 2 always produce clean integer results.

Most scientific calculators only have buttons for log₁₀ and ln. To compute log_b(x) for any other base b, you use the change-of-base formula:

Derivation: let y = log_b(x), so b^y = x. Take the natural log of both sides: ln(b^y) = ln(x). Apply the power rule: y·ln(b) = ln(x). Divide both sides by ln(b): y = ln(x)/ln(b). That is all — the change-of-base formula is a direct consequence of the power rule for logarithms.

Example: log₃(81). Since ln(81) ≈ 4.3944 and ln(3) ≈ 1.0986, log₃(81) = 4.3944 / 1.0986 = 4. Verification: 3⁴ = 81 ✓. This calculator shows both the ln-form and the log₁₀-form of the derivation so you can verify using whichever reference base your calculator provides.