Logarithm Calculator
Calculate logarithms with any base, natural logarithms (ln), common logarithms (log_1₀), and antilogarithms. Perfect for science, engineering, and mathematical applications.
What is a logarithm and how does it relate to exponents?
A logarithm is the inverse of exponentiation. If aˣ = y, then logₐ(y) = x. Read as "log base a of y equals x". Example: 2^3 = 8, so log_2(8) = 3. The logarithm answers "what power do I raise the base to, to get this number?" Common bases: 10 (common log), e~=2.718 (natural log), 2 (binary). Key relationship: a^(logₐ(x)) = x and logₐ(aˣ) = x.
What is the difference between ln, log, and log_1₀?
ln(x) = natural logarithm = logₑ(x), base e ~= 2.71828. Used in calculus, growth/decay, continuous compounding. log(x) notation varies: in math/science = log_1₀(x) (common log), in computer science/advanced math = ln(x). log_1₀(x) = common logarithm, base 10. Used for pH, decibels, Richter scale. Conversion: ln(x) = log_1₀(x) / log_1₀(e) ~= 2.303 * log_1₀(x). Always check context for "log" meaning.
What are the main logarithm properties and laws?
Product Rule: log(xy) = log(x) + log(y). Example: log(100) = log(10*10) = log(10) + log(10) = 1+1 = 2. Quotient Rule: log(x/y) = log(x) - log(y). Power Rule: log(xⁿ) = n·log(x). Example: log(1000) = log(10^3) = 3·log(10) = 3. Change of Base: logₐ(x) = log(x)/log(a) or ln(x)/ln(a). Identity: logₐ(a) = 1, logₐ(1) = 0. Inverse: a^(logₐ(x)) = x.
How do you calculate logarithms without a calculator?
For simple cases, recognize patterns: log_1₀(1000) = 3 because 10^3=1000. log_2(32) = 5 because 2^5=32. For other values: 1) Use log tables (historical method), 2) Change of base to known values: log₅(25) = ln(25)/ln(5) = 3.219/1.609 = 2. 3) Use properties: log_2(24) = log_2(8*3) = log_2(8) + log_2(3) = 3 + 1.585 = 4.585. 4) Approximate: log_1₀(50) is between log_1₀(10)=1 and log_1₀(100)=2, closer to 2.
What is an antilogarithm and when is it used?
Antilog is the inverse of logarithm. If log_1₀(x) = 2, then antilog_1₀(2) = x = 100. It's exponentiation: antilogₐ(x) = aˣ. Example: If ln(y) = 3.5, then y = antilog_e(3.5) = e^3.5 ~= 33.12. Used when: 1) Solving log equations, 2) Decoding log-transformed data (pH->[H⁺], decibels->intensity), 3) Logarithmic scales (earthquake magnitude->energy). Think: log compresses, antilog expands.
Why can't you take the logarithm of zero or negative numbers?
For real logarithms, log(0) and log(negative) are undefined because no real power makes positive base equal zero or negative. Example: 10ˣ is always positive, never 0 or negative, so log_1₀(0) and log_1₀(-5) don't exist in real numbers. However, in complex numbers, they exist: ln(-1) = iPI (using Euler's formula). Domain restrictions: x > 0 for logₐ(x), and a > 0, a != 1. log(0⁻) -> -infinity.
What are common real-world applications of logarithms?
Applications everywhere: 1) pH scale: pH = -log[H⁺], each unit = 10* change. pH 3 is 10* more acidic than pH 4. 2) Richter scale: each number = 31.6* more energy. 3) Decibels: dB = 10·log(I/I₀), sound intensity. 4) Finance: compound interest, time to double money. 5) Data science: log transformation for skewed data. 6) Computer science: algorithm complexity (binary search = O(log n)). 7) Biology: population growth, allometric scaling.
How do you solve logarithmic equations?
Methods: 1) Convert to exponential form: log_2(x) = 5 -> x = 2^5 = 32. 2) Use log properties: log(x) + log(x+3) = 1 -> log(x(x+3)) = 1 -> x(x+3) = 10 -> x^2 + 3x - 10 = 0 -> x = 2. 3) Same base method: log_2(x) = log_2(8) -> x = 8. Always check: solutions must make argument > 0. False solution example: log(x-2) = 1 gives x=12 ✓ or x=-2 ✗ (makes log(-4), invalid).
What is the change of base formula and why is it useful?
Formula: logₐ(x) = logᵦ(x) / logᵦ(a) for any valid base b. Most common: logₐ(x) = ln(x)/ln(a) or log_1₀(x)/log_1₀(a). Why useful? Calculators only have ln and log_1₀ buttons. To find log₅(30): = ln(30)/ln(5) = 3.401/1.609 = 2.113. Or use log_1₀: = log_1₀(30)/log_1₀(5) = 1.477/0.699 = 2.113. Also useful for comparing logs with different bases.
How do logarithmic and exponential functions relate graphically?
They're reflections across y=x line (inverse functions). Exponential y=aˣ: passes through (0,1), increases/decreases rapidly, horizontal asymptote at y=0, domain: all x, range: y>0. Logarithmic y=logₐ(x): passes through (1,0), increases/decreases slowly, vertical asymptote at x=0, domain: x>0, range: all y. For a>1: both increasing. For 0<a<1: both decreasing. Key point swap: (0,1) exponential ↔ (1,0) logarithm.