A logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.
In Spanish
logaritmo
Also called
log, log base b, common logarithm, inverse of an exponential
The base stays the base, the exponent becomes the answer of the log, and the result moves inside the log.
Formal definition and notation
For a base b > 0 with b ≠ 1 and a numberx > 0, the logarithm logbx is the unique real numbery such that by = x. The function f(x) = logbx is the inverse of the exponential function g(x) = bx: logbbx = x and blogbx = x. The common logarithm logx has base 10, and the natural logarithmlnx has base e ≈ 2.718.
Symbol
Read it as
Meaning
logbx
log base b of x
the exponent that turns b into x
logx
log of x
the common logarithm, base 10 (on calculators, the LOG key)
lnx
natural log of x, or "lon x"
the natural logarithm, base e (the LN key)
Key facts
A logarithm is an exponent. To evaluatelog3 81, ask “3 to what power is 81?” The answer is 4.
You cannot take the log of 0 or a negative number: no power of a positive base is 0 or negative.
Logs of numbers between 0 and 1 are negative: log 0.01 = −2 because 10−2 = 0.01.
Logs turn multiplication into addition. Each time x is multiplied by 10, logx goes up by 1, which is why the pH, decibel and Richter scales use logs.
To solve an exponential equation such as 2t = 50, take a log: t = log2 50 = log 50log 2 ≈ 5.64.
Logarithm definition and rules
Logarithm definition and rules
logbx = y ⇔ by = x
product rule
logb (MN) = logbM + logbN
quotient rule
logb ( MN ) = logbM − logbN
power rule
logb (Mk) = klogbM
change of base
logbx = logxlogb, which lets a base-10 calculator find any log
Conditions: b > 0, b ≠ 1, and the input must be positive. Two values never change: logb 1 = 0 and logbb = 1.
Try it: find the missing exponent
Interactive
A log asks for the exponent. Keep the base at 2 and change the exponent until the result is 64: that exponent is log2 64. Try negative exponents to see logs of fractions.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
A square root finds the base: √32 asks what number squared is 32. A log finds the exponent.
Quick check: 3 questions
Question 1 of 3
What is log4 64?
Show answer and explanation
Answer: D.43 = 64, so the log is 3. 16 divides 64 by 4, 60 subtracts, and 4 is the base.
Question 2 of 3
Which equation means the same as log7 49 = 2?
Show answer and explanation
Answer: C. Base 7, exponent 2 (the log’s value), result 49: 72 = 49. The other choices put the base or the answer in the wrong place.
Question 3 of 3
What is log 0.01?
Show answer and explanation
Answer: B. No base means base 10, and 10−2 = 1100 = 0.01, so the log is −2. 2 forgets the negative exponent; the decimals treat the log as multiplication.
Logarithm: frequently asked questions
What is a logarithm in simple terms?
A logarithm is a question about exponents: what power of the base gives this number? log10 1000 asks “10 to what power is 1000?” and the answer is 3. Logarithms are how you solve for an exponent, the same way division solves for a missing factor.
What does log mean with no base?
In school math and on calculators, logx with no base means the common logarithm, base 10, so log 100 = 2. The natural logarithm uses base e and is written lnx. In some college and computer-science books, an unmarked log can mean base e or base 2, so check the book’s convention.
Why can you not take the log of a negative number?
Because a positive base raised to any real power is always positive. No exponent makes 10y = −5 or 10y = 0, so log (−5) and log 0 are undefined in the real numbers. The input of a log, called its argument, must be greater than 0.
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