Logarithm Calculator
Solve log(x), ln(x) & log Base b(x)
Calculate logarithms in base 10, natural log (ln), or any custom base instantly — with full step-by-step working using the change of base formula. Built for TN Board, CBSE, JEE and NEET students.
Enter your logarithm
How to Calculate a Logarithm
A logarithm answers the question: "to what power must the base be raised to get this number?" The expression log_b(x) = y means by = x. The two most common logarithms are the common logarithm (base 10, written log₁₀ or just log) and the natural logarithm (base e ≈ 2.71828, written ln).
log_b(x) = ln(x) / ln(b)
This is called the change of base formula, and it's how this calculator finds a logarithm in any base — by converting it to natural logs first, then dividing. It's one of the most frequently tested identities in Class 11–12 Maths, JEE and NEET.
A useful check after calculating any logarithm is the antilog — raising the base back to the power of your answer should give you back the original value (x). This calculator shows that check automatically so you can verify your working.
Some quick facts worth remembering: log_b(1) = 0 for any base, log_b(b) = 1, and the logarithm of a negative number or zero is undefined in the real number system — this calculator will flag that for you.
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"log" usually refers to the common logarithm, base 10 (log₁₀), while "ln" refers to the natural logarithm, base e (≈2.71828). Both follow the same rules, but they're used in different contexts — log₁₀ is common in engineering and pH/decibel scales, while ln appears throughout calculus, growth/decay problems, and higher mathematics.
Because no real power of a positive base can ever produce a negative result — raising any positive base to any real exponent always gives a positive number. Logarithms of negative numbers do exist in the complex number system, but that's beyond the standard real-number logarithm covered in school and most competitive exams.
log_b(x) equals ln(x) divided by ln(b) — or equivalently, log₁₀(x) divided by log₁₀(b). This lets you calculate a logarithm in any base using only the log or ln function on a basic calculator, which typically only has buttons for base 10 and base e.
An antilog reverses a logarithm — if log_b(x) = y, then the antilog of y (in base b) gives back x, calculated as b raised to the power y. It's commonly used to verify a logarithm answer, or to solve equations where the logarithm of an unknown value is given.
Logarithms appear in the Richter scale for earthquakes, the pH scale in Chemistry, decibel sound measurement, compound interest and population growth formulas, and computer science algorithm complexity. In exams, they're core to Class 11-12 Maths (logarithmic and exponential equations), Chemistry (pH calculations), and are frequently tested in JEE and NEET.