Free online Log Calculator & Antilog Suite. Solve log_b(x), natural logs ln(x), common logs log10(x), binary logs log2(x), change of base, and interactive curves.
Logarithms are the inverse operation of exponentiation. A logarithm answers the question: "What exponent must a base be raised to in order to produce a given number?"
This Log Calculator evaluates logarithms with custom bases as well as natural, common and binary logarithms. It can also calculate antilogarithms, solve logarithmic equations for different variables, demonstrate the change-of-base formula, and show the calculation step by step.
For a real logarithm log_b(x) = y, the equivalent exponential statement is b^y = x. The calculator uses this fundamental relationship throughout its logarithm, antilogarithm and three-variable solving tools.
A logarithm is the inverse of an exponential operation. The statement log_b(x) = y means exactly the same thing as:
Here b is the base, x is the argument, and y is the logarithm. For example, log_2(8) = 3 because 2^3 = 8. Similarly, log_10(100) = 2 because 10^2 = 100.
Thinking of a logarithm as "the exponent required to produce a number" makes the connection between logarithms and exponential equations much easier to understand.
Enter the logarithm base and the argument in the main logarithm calculator. For log_b(x), enter:
The calculator then provides:
The other sections handle inverse and equation-solving tasks. Use the Antilogarithm & Exponential Solver when you know a base and exponent and want to calculate the resulting value. Use the 3-Variable Logarithm Equation Solver when the relationship is log_b(x) = y and you want to solve for y, x, or b.
For a real logarithm log_b(x), the standard domain conditions are:
These restrictions are part of the definition of a real logarithm. The argument x must be positive. A logarithm of zero is undefined in the real numbers, and a logarithm of a negative argument is not a real-valued logarithm. The base must also be positive and cannot equal 1. If the base were 1, then 1^y would always equal 1, so it could not uniquely produce arbitrary positive arguments.
The calculator validates these domain conditions rather than silently forcing invalid inputs into a numerical formula.
| Quantity | Requirement | Mathematical Justification |
|---|---|---|
| Base b | b > 0 | Standard real logarithm; negative bases cannot sustain arbitrary real powers without complex numbers. |
| Base b | b ≠ 1 | Base 1 cannot uniquely generate x, because 1ʸ = 1 for all real exponents y. |
| Argument x | x > 0 | Real logarithm domain; raising a positive base to any real exponent always yields a positive result. |
A logarithm does not have to use base 10 or base e. For example, log_5(125) = 3 because 5^3 = 125. You can also evaluate log_3(81) = 4 and log_2(64) = 6.
The ability to enter a custom base is one of the main advantages of this calculator over a standard calculator that may expose only ln and log buttons.
When a calculator does not directly provide log base b, the change-of-base formula converts it into a quotient of logarithms that are easier to evaluate. The formula is:
It can also be written with any common reference base where the same base is used in the numerator and denominator. For example, log_5(36) can be evaluated as ln(36) / ln(5) ≈ 3.583519 / 1.609438 ≈ 2.2266.
The calculator uses this relationship for arbitrary-base logarithms and shows the intermediate natural-log calculations in its step-by-step derivation.
"Logarithms and exponentials describe the same relationship from opposite directions."
When your problem involves powers or exponent rules rather than logarithms, the Exponent Calculator provides a dedicated workflow for evaluating powers, fractional exponents and exponent laws.
Consider log_10(100). The logarithmic definition tells us that 10^y = 100. Since 10^2 = 100, the result is:
The calculator also verifies this using change of base:
The final step converts the logarithmic answer back to exponential form: 10^2 = 100. This is a useful check because logarithms and exponentials are inverse operations.
A custom base does not have to be a familiar value such as 2, 10 or e. Consider:
Therefore, log_1049(105) ≈ 0.6690961665. The calculator's interactive graph uses the same active base and marks the current argument point, so the visualization remains connected to the calculation rather than displaying a fixed demonstration curve.
The natural logarithm is the logarithm with base e: ln(x) = log_e(x), where Euler's constant e ≈ 2.718281828459... Examples include ln(1) = 0, ln(e) = 1, and ln(e^2) = 2.
Natural logarithms are especially common in calculus, differential equations, continuous growth and decay, and scientific models involving exponential functions. The calculator reports ln(x) alongside arbitrary-base logarithms so that you can compare different logarithmic representations of the same argument.
The common logarithm uses base 10: log_10(x). Examples: log_10(10) = 1, log_10(100) = 2, log_10(1000) = 3, and log_10(0.01) = -2.
The last example illustrates an important point: a logarithm can be negative even though its argument is positive. A negative logarithm means the argument lies between 0 and 1 when the base is greater than 1.
The binary logarithm uses base 2: log_2(x). It appears frequently in computer science, information theory and algorithms. Examples include log_2(2) = 1, log_2(8) = 3, log_2(64) = 6, and log_2(1024) = 10. Binary logarithms are especially useful when quantities grow or shrink by factors of two.
| Type | Base | Common Use | Canonical Example |
|---|---|---|---|
| Common log | 10 | General / scientific notation contexts & engineering magnitude | log₁₀(100) = 2 |
| Natural log | e ≈ 2.718 | Calculus, continuous models, physics & differential equations | ln(e²) = 2 |
| Binary log | 2 | Computing, binary search trees, bit depth & information theory | log₂(64) = 6 |
| Custom-base log | b | General logarithmic calculations & arbitrary base change proofs | log₅(125) = 3 |
A logarithm can be negative without being invalid. For a base greater than 1, log_b(x) < 0 whenever 0 < x < 1. For example, log_10(0.01) = -2 because 10^-2 = 0.01. The sign of the logarithm therefore depends on the argument relative to 1 and the base being used.
An antilogarithm reverses a logarithm. If log_b(x) = y, then x = b^y. Therefore, antilog_b(y) = b^y.
For example, antilog_10(2) = 10^2 = 100. The calculator's Antilogarithm & Exponential Solver performs this inverse calculation and also shows the exponential relationship used to obtain the result.
Suppose y = 2 and b = 10. Then antilog_10(2) = 10^2 = 100. We can verify the answer with a logarithm: log_10(100) = 2. This inverse relationship is one of the most important connections between logarithms and exponentials.
The equation log_b(x) = y can be rewritten as b^y = x. This means any one of the three quantities can be solved when the other two are known. The calculator provides three target modes:
Given base b and argument x, evaluate y = log_b(x). Example: log₂(64) = 6.
Given base b and exponent y, evaluate x = b^y. Example: 2⁶ = 64.
Given argument x and exponent y, solve b = x^(1/y) when real roots exist.
Given b = 2 and x = 64, we have log_2(64) = y. Since 2^6 = 64, the answer is y = 6. The calculator verifies the result by converting the logarithmic equation back into exponential form.
Given b = 2 and y = 6, start with log_2(x) = 6. Convert to exponential form: x = 2^6 = 64. The calculator displays this relationship as part of its step-by-step equation solution.
Given x = 64 and y = 6, we have b^6 = 64. Taking the sixth root gives b = 64^(1/6) = 2. The result can be checked by substitution: 2^6 = 64.
For negative arguments and odd integer exponents, a negative real base can also be mathematically valid. For example, b^3 = -8 gives b = -2. The calculator distinguishes valid real-domain cases from cases with no real solution.
When the main task is evaluating a power or fractional exponent rather than solving a logarithmic equation, use the Exponent Calculator for the direct exponentiation workflow.
The graph of y = log_b(x) provides a visual interpretation of the logarithm. For a valid real base b > 0 and b ≠ 1, the graph has several important characteristics:
x > 0.(-∞, +∞).x = 0.(1, 0).(b, 1).When b > 1, the logarithmic curve is strictly increasing. When 0 < b < 1, the curve is strictly decreasing. The calculator's graph updates its base dynamically rather than remaining a fixed base-2 illustration.
The base determines the shape and direction of a logarithmic curve. For b = 2, the graph passes through (1, 0) and (2, 1). For b = 10, the corresponding base point becomes (10, 1). For 0 < b < 1, the curve is decreasing instead of increasing.
The calculator's base slider is synchronized with the active logarithm. Moving the slider changes the graph's base, while valid changes to the main base input update the graph as well. This synchronization is important because a graph showing log_2(x) while the calculation uses another base would be mathematically misleading.
In addition to the universal point (1, 0) and the base point (b, 1), the calculator can display the current calculation point. If y = log_b(x), then the point is (x, y). For example, with b = 1049 and x = 105, the point is approximately (105, 0.6691). This connects the numerical result directly to the plotted logarithmic curve.
The logarithmic function is defined only for positive arguments. As x approaches zero from the positive side, the logarithm decreases without bound when b > 1. Therefore, x = 0 acts as a vertical asymptote. The graph does not cross the y-axis because zero is outside the domain of the real logarithm.
For appropriate positive real arguments and valid base b, logarithms obey several important algebraic properties:
These rules make it possible to expand products, condense logarithmic expressions and solve equations more efficiently. The domain requirements still apply. For real logarithms, the arguments of the individual logarithms must be positive.
Consider log_10(10 × 100). Using the product rule: log_10(10 × 100) = log_10(10) + log_10(100) = 1 + 2 = 3. Since 10 × 100 = 1000, we can verify log_10(1000) = 3.
Consider log_10(1000/10). Using the quotient rule: log_10(1000/10) = log_10(1000) - log_10(10) = 3 - 1 = 2. And 1000/10 = 100, so log_10(100) = 2.
For log_b(x^k), the power can move in front: log_b(x^k) = k log_b(x). For example: log_10(100²) = 2 log_10(100) = 2 × 2 = 4, which agrees with 10^4 = 10000.
"Logarithm" describes the general operation, while the notation often identifies a particular base. Common logarithm: log_10(x). Natural logarithm: ln(x) = log_e(x). Binary logarithm: log_2(x).
The base changes the numerical value of the logarithm, but all three functions share the same inverse relationship with their corresponding exponential functions. For example: log_10(100) = 2, ln(e²) = 2, and log_2(64) = 6.
For any valid logarithm base, log_b(1) = 0 because b^0 = 1. Therefore the logarithm of 1 is zero for every base satisfying b > 0 and b ≠ 1. The graph of every parent logarithmic function therefore passes through (1, 0).
Yes. For a base greater than 1, arguments between 0 and 1 produce negative logarithms. For example, log_10(0.1) = -1 because 10^-1 = 0.1, and log_10(0.01) = -2 because 10^-2 = 0.01. A negative result does not mean that the argument was invalid.
A logarithm requires b > 0 and b ≠ 1. The reason 1 is excluded is that 1^y = 1 for every real y. Therefore, the equation 1^y = x cannot uniquely produce arbitrary positive values of x. A logarithm therefore cannot use 1 as its base.
In the real-number definition of a logarithm, the base must be positive. The exponential function b^y with a positive base is defined for every real y and produces positive outputs. A negative base does not have a corresponding real-valued exponential function for every real exponent, so it is excluded from the standard real logarithm definition. The calculator therefore rejects negative logarithm bases in its real-domain logarithm calculation.
There is no real number y satisfying b^y = 0 when b > 0. A positive base raised to any finite real power remains strictly positive. Therefore, log_b(0) is undefined in the real numbers. Graphically, this is connected to the vertical asymptote x = 0.
For a real logarithm, b^y = x with positive base b produces a positive result. Therefore it cannot produce a negative x. For example, log_2(-8) has no real-valued result. The calculator identifies this as a real-domain error instead of silently returning an incorrect numerical value.
The exponential function y = b^x and logarithmic function y = log_b(x) are inverse functions for valid bases. This means:
within their respective domains. Their graphs are reflections across y = x when the corresponding functions are plotted together. This inverse relationship explains why logarithms are useful for solving equations in which the unknown appears in an exponent.
Suppose 2^x = 20. Because 20 is not an obvious power of 2, logarithms can isolate x. Take logarithms: log(2^x) = log(20). Apply the power rule: x · log(2) = log(20). Therefore:
The same result can be obtained using natural logarithms. This is the main reason logarithms are important for solving exponential equations.
When a logarithm is being used to isolate an exponent, a dedicated Scientific Calculator can also be useful for checking the resulting numerical expression and trigonometric ratios.
Logarithms are closely connected to powers of ten. For example, log_10(1000) = 3 because 10^3 = 1000. Scientific notation uses this same power-of-ten structure to represent very large and very small numbers. For example, 5,400,000 = 5.4 × 10^6 and 0.0000054 = 5.4 × 10^-6. A logarithm can therefore be interpreted as a way of measuring the exponent associated with a particular base.
Explore the Scientific Notation Calculator & Converter for direct base-10 exponential form and engineering scale formatting.
Logarithmic relationships appear whenever quantities span a wide range or change multiplicatively rather than additively.
Decibel formulas use logarithms to express ratios of physical quantities over a convenient scale. For power or intensity ratios, a common form is L = 10 log₁₀(P/P₀). The exact formula depends on the physical quantity being measured, so power, intensity and pressure-amplitude formulas should not be mixed.
pH is logarithmically related to hydrogen-ion activity, commonly represented in introductory contexts by pH = -log₁₀[H⁺]. This means a change of one pH unit corresponds to a tenfold change in the hydrogen-ion concentration under the simplified concentration relationship.
Binary logarithms appear naturally in algorithms such as binary search: O(log₂ n). The logarithm base does not change the asymptotic Big-O class because changing the logarithm base only introduces a constant multiplying factor.
Logarithmic scales can make extremely large or small ratios easier to describe and compare. The important point is that the logarithmic formula must match the physical quantity and measurement convention being used.
Earthquake magnitude scales are logarithmic, but different magnitude measures and physical quantities do not all use the same numerical relationship. For this reason, statements such as "one magnitude unit means X times more energy" should always identify what physical quantity and magnitude scale are being discussed.
A logarithmic scale generally means that equal changes in the reported magnitude correspond to multiplicative changes in the underlying measured quantity. Do not interpret a logarithmic scale as though each unit were a fixed additive increase in the physical quantity.
The simplest verification is to return to exponential form. If log_b(x) = y, then check b^y = x.
log_2(64) = 6: check 2^6 = 64.log_10(0.01) = -2: check 10^-2 = 0.01.log_1049(105) ≈ 0.6690961665: check 1049^0.6690961665 ≈ 105.The calculator also performs this kind of exponential-form verification in its step-by-step derivation.
Some logarithms have simple exact values like log_2(64) = 6 and log_10(100) = 2. But many logarithms are irrational numbers and must be represented numerically, such as log_2(10) ≈ 3.32192809489. A displayed decimal is therefore an approximation of the exact mathematical quantity. The calculator retains higher precision internally and formats the displayed result for readability.
These examples cover common-base, natural, binary, custom-base and inverse logarithmic calculations represented by the calculator.
The graph is not a decorative image. It represents y = log_b(x) using the calculator's current base. Three important points help connect the formula with the graph: (1, 0), (b, 1), and (x, log_b(x)). The first point is universal. The second depends directly on the selected base. The third corresponds to the user's current argument. Changing the base therefore changes both the numerical logarithm and the geometry of the displayed curve.
The calculator includes tools for preserving calculations after they have been completed:
The core mathematical concepts used by this calculator are standard properties of real logarithmic and exponential functions. The definition log_b(x) = y ⇔ b^y = x, domain conditions b > 0, b ≠ 1, x > 0, change-of-base relationship log_b(x) = ln(x)/ln(b), and graph characteristics are covered extensively in standard textbooks.
Evaluate powers, exponential bases, fractional exponents and algebraic exponent rules with full derivations.
Advanced numerical calculator supporting trigonometric, inverse trigonometric, exponential and multi-step expressions.
Calculate square roots, cube roots and general nth roots while breaking down rads into prime factorizations.
Convert standard decimal numbers to power-of-ten scientific notation, engineering notation and e-notation formats.
This calculator applies standard real-number logarithm and exponential relationships to the values entered by the user. A mathematically valid calculation does not by itself determine whether a particular real-world model or equation is appropriate.
For important academic, scientific or professional work, verify the result against the assumptions, domain, units and definitions appropriate to the problem.