A logarithm answers an exponent question: to what power must a chosen base be raised to produce a given number? This calculator evaluates logarithms for algebra, science, computing, acoustics, chemistry, and exponential-growth problems where direct powers grow too large to compare comfortably. Instead of working with the full number, you work with its exponent relative to a base, making scale and multiplication easier to reason about.
Quick Answer
If bʸ = x, then log_b(x) = y. Here b is the base, x is the positive number, and y is the logarithm.
Logarithms Turn Powers Into Ordinary Values
The base determines the counting system. Base 10 asks how many powers of ten build the number: log₁₀(1000) = 3 because 10³ = 1000. Base 2 is common in computing: log₂(32) = 5 because 2⁵ = 32. The natural logarithm uses base e and is written ln(x), appearing throughout continuous growth, decay, and calculus. These are different logarithms of the same number because each measures it against a different base.
The input to a real-valued logarithm must be greater than zero. Zero and negative numbers do not have real logarithms under the ordinary definition because a positive base raised to a real power remains positive. The base must be positive and cannot equal 1; base 1 produces only 1 and cannot generate a range of values. A logarithm itself, however, can be negative: log₁₀(0.01) = −2 because 10⁻² = 0.01.
Several identities explain how calculators and people simplify expressions. A product becomes a sum of logarithms, log_b(xy) = log_b(x) + log_b(y). A quotient becomes a difference, and a power can move in front: log_b(xⁿ) = n log_b(x). These rules apply only when the same base is used and the inputs satisfy the domain conditions. The inverse operation is handled by the Antilog Calculator.
How to Use the Log Calculator
- Identify the number whose logarithm you need.
- Identify the base specified by the problem or application.
- Check that the number is positive and the base is positive and not equal to 1.
- Enter the number and base.
- Read the logarithm, retaining enough decimal places for the next step of the problem.
- Verify by raising the base to the result and checking that it returns the original number.
Two Logarithms Checked by Their Powers
Find log₂(32). Ask what exponent turns 2 into 32. The powers are 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, and 2⁵ = 32. Therefore log₂(32) = 5. No approximation is needed because 32 is an exact power of 2.
Most values fall between exact powers. For example, log₁₀(250) is about 2.39794 because 10²·³⁹⁷⁹⁴ is approximately 250. Its integer part tells us the number lies between 10² and 10³, while the fractional part locates it more precisely within that range. This is why logarithms compress scale: every increase of 1 in a base-10 logarithm represents a tenfold increase in the original number.
- log₁₀(100) = 2.
- log₅(25) = 2.
- log₃(1) = 0, because any valid base to the power 0 equals 1.
- log₂(1/8) = −3, because 2⁻³ = 1/8.
Base Confusion and Invalid Inputs
Many mistakes begin with an unstated base. On some calculators, "log" means base 10; in advanced mathematics, it may mean natural logarithm depending on context. A problem that expects ln(20) will not accept log₁₀(20), even though both are legitimate values. Rewrite the notation explicitly before calculating, and if a result will be shared, name the base in the answer.
Do not split a logarithm across addition inside its input. In general, log(x + y) is not log(x) + log(y); the product rule applies to multiplication, not addition. Likewise, log(x/y) is a difference of logarithms, not ordinarily the quotient log(x) / log(y). The change-of-base formula is log_b(x) = ln(x) / ln(b), and there the division occurs between two logarithms after the inputs are evaluated.
Scientific notation can help anticipate the size of a base-10 logarithm, especially for very large or tiny measurements. The Scientific Notation Calculator is useful when rewriting such inputs. Additional algebra tools are collected in Mathematics Calculators.
Changing Base When Only One Logarithm Is Available
The change-of-base identity lets any logarithm be evaluated from a natural or common logarithm: log_b(x) = log(x) / log(b), provided the same logarithm is used in the numerator and denominator. For instance, log₂(20) equals ln(20) / ln(2), approximately 4.32193. That result makes sense because 2⁴ = 16 and 2⁵ = 32, so the exponent for 20 must fall between 4 and 5, closer to 4.
Keep the division in the correct order. The desired base goes into the denominator through its logarithm; reversing the ratio gives log₂₀(2), a much smaller number. Bracketing the answer between consecutive integer powers provides an independent check. If log₃(50) is calculated, note that 3³ = 27 and 3⁴ = 81, so the answer must lie between 3 and 4. A displayed result outside that interval means the base, input, or change-of-base arrangement needs attention.
Logarithms also provide a disciplined way to compare quantities that span many orders of magnitude. On a base-10 scale, values of 3, 5, and 7 correspond to original quantities separated by factors of 100 at each step, not by equal additions. This is why a difference of 0.3 in a logarithm represents roughly a doubling, because 10⁰·³ is about 1.995. When reading a result, ask whether the problem asks for the exponent itself or expects you to return to the original scale. Handing in the logarithm when an antilog was required is a common final-step error.
Graphing gives another perspective. The logarithm curve rises quickly at first for a base greater than one and then flattens, reflecting the fact that equal multiplications of the input add equal amounts to the output. Doubling an input always adds log_b(2), regardless of whether the input is small or enormous. That property underlies slide-rule arithmetic and many scientific scales. It also means a tiny input error near zero can move the logarithm a great deal, so measurements close to zero need particular care.
Verify Every Logarithm by Undoing It
The clearest check is inversion. If the calculator says log_b(x) = y, raise b to y and compare it with x. Small differences should be explainable by rounding; a completely different value usually means the base, notation, or input was misunderstood.