Log Calculator: Solve Logarithmic Expressions Instantly
Logarithms are essential in mathematics, science, and engineering for solving exponential equations, measuring pH levels, calculating earthquake magnitudes, and analyzing data that spans many orders of magnitude. Our Log Calculator computes logarithms in any base, natural logarithms (ln), and common logarithms (log10), with step-by-step solutions showing how the result is derived. Use our Logarithm Calculator to estimate your costs.
Understanding Logarithms
A logarithm answers the question: to what exponent must a base be raised to produce a given number? If b^x = y, then log_b(y) = x. For example, log_2(8) = 3 because 2³ = 8. The common logarithm uses base 10, and the natural logarithm uses base e (approximately 2.71828).
Logarithms convert multiplication into addition: log(x × y) = log(x) + log(y). They convert division into subtraction: log(x / y) = log(x) - log(y). And they convert exponents into multiplication: log(x^n) = n × log(x). These properties make logarithms invaluable for simplifying complex calculations.
Using the Log Calculator
Enter the number you want to take the logarithm of, and the base. The calculator supports any positive base other than 1. Select common log (base 10), natural log (base e), or custom base. The calculator shows the result, the logarithmic equation in exponential form, and how the logarithm properties were applied.
You can also solve logarithmic equations: enter an expression like log(x) + log(x+2) = 3 and the calculator solves for x using logarithmic properties.
Applications of Logarithms
pH Scale
pH = -log₁₀[H+], where [H+] is the hydrogen ion concentration. A solution with pH 7 has 10 times more hydrogen ions than pH 8. Each whole pH unit represents a tenfold change in acidity.
Richter Scale
Earthquake magnitude uses a logarithmic scale. A magnitude 6 earthquake releases approximately 32 times more energy than magnitude 5, and 1,000 times more than magnitude 4.
Decibels
Sound intensity is measured in decibels on a logarithmic scale. A 10 dB increase represents a tenfold increase in sound intensity. 90 dB (heavy traffic) is 1,000 times more intense than 60 dB (normal conversation).
Exponential Growth and Decay
Population growth, compound interest, radioactive decay, and drug metabolism all follow exponential patterns. Logarithms are used to linearize these relationships for analysis and prediction.
Logarithmic Properties
- Product rule: log_b(xy) = log_b(x) + log_b(y)
- Quotient rule: log_b(x/y) = log_b(x) - log_b(y)
- Power rule: log_b(x^n) = n × log_b(x)
- Change of base: log_b(x) = log_a(x) / log_a(b) — useful when your calculator only has log₁₀ or ln
- Inverse property: b^(log_b(x)) = x and log_b(b^x) = x
- Log of 1: log_b(1) = 0 for any base b
- Log of base: log_b(b) = 1
Real-World Example
A savings account with $10,000 earns 5% annual interest compounded continuously. How long until the balance reaches $15,000?
- Formula: A = P × e^(rt), so $15,000 = $10,000 × e^(0.05t)
- Divide both sides: 1.5 = e^(0.05t)
- Take natural log: ln(1.5) = 0.05t
- Solve: t = ln(1.5) / 0.05 = 0.4055 / 0.05 = 8.11 years
Checking: log₁₀(1,000) = 3 because 10³ = 1,000. log₂(32) = 5 because 2⁵ = 32. These relationships are the foundation of logarithmic thinking.
How to Read the Results and Common Pitfalls
The calculator returns five numbers at once for any input: the logarithm in your chosen base, the natural log ln, the common log log₁₀, the binary log log₂, and the antilog, which is the base raised to the result. Seeing them side by side makes the relationships between scales concrete. For input 100 and base 10, you get log₁₀(100) = 2, ln(100) = 4.605170, log₂(100) = 6.643856, and the antilog confirms 10² = 100.
Try the same idea with base 2. Enter number 32 and base 2 and the result is 5, because 2⁵ = 32. The ln column shows 3.465736 and the log₁₀ column shows 1.505150; notice that ln(32) ÷ ln(2) = 3.465736 ÷ 0.693147 = 5, which is exactly the change-of-base formula applied internally. That is why the tool can return a base-2 or base-3 logarithm even though many handheld calculators only carry log₁₀ and ln buttons.
Common mistakes to avoid when using the tool:
- Entering a non-positive number. The logarithm is only defined for positive inputs. Enter 0 or a negative value and there is no real answer, so the calculator flags the input as invalid. Double-check that a concentration, pressure, or count is entered as a positive value.
- Using base 1. A base of 1 is undefined because 1 raised to any power is still 1, so it can never produce a value other than 1. The calculator rejects base 1 for the same reason textbooks do.
- Confusing log and ln in pH work. The pH formula uses base 10 by definition: pH = -log₁₀[H⁺]. Substituting ln instead would shift the answer by a factor of about 2.303, turning a neutral pH of 7 into roughly 16.1.
- Misreading decimal outputs. A negative result like -4.49485 for a small input is expected, not an error: 10 raised to a negative exponent produces a fraction, so the log of 0.000032 is negative.
Once you are comfortable reading all five outputs, you can verify every calculation by eye: multiply the result by ln(base) and it should equal ln(number). That cross-check is the fastest way to catch a keystroke mistake before trusting the number in a homework answer or lab report.
Start Calculating
Use our Log Calculator below to solve logarithmic expressions and equations. Also check our Scientific Calculator for advanced calculations and our Standard Deviation Calculator for statistical analysis.
Related Calculators
Use our Log Calculator Guide together with related tools such as Area Calculator, Combination Calculator, Derivative Calculator to plan more accurately. Each calculator runs instantly in your browser with step-by-step guidance.
Written by the CalcMaster Pro Editorial Team — financial, health, and DIY tools reviewed for accuracy. All calculators run on standard, widely accepted formulas. Always confirm final numbers with a qualified professional for decisions that require official figures.