Log Calculator (Logarithm)
Please provide any two values to calculate the third in the logarithm equation logb x = y. It can accept "e" as a base input.
What is Log?
The logarithm, or log, is the inverse of the mathematical operation of exponentiation. This means that the log of a number is the power that a fixed base has to be raised to in order to yield the number. Conventionally, log implies that base 10 is being used, though the base can technically be anything. When the base is e, ln is usually written rather than loge. log2, the binary logarithm, is another base typically used with logarithms. If, for example:
x = by ; then y = logb x ; where b is the base
Each of the mentioned bases is typically used in different applications. Base 10 is commonly used in science and engineering, base e in math and physics, and base 2 in computer science.
Basic Log Rules
When the argument of a logarithm is the product of two numerals, the logarithm can be re-written as the sum of the logarithm of each of the numerals.
logb(x × y) = logb x + logb y
EX: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1
When the argument of a logarithm is a fraction, the logarithm can be re-written as the subtraction of the logarithm of the numerator minus the logarithm of the denominator.
logb(x / y) = logb x − logb y
EX: log(10 / 2) = log(10) − log(2) = 1 − 0.301 = 0.699
If there is an exponent in the argument of a logarithm, the exponent can be pulled out of the logarithm and multiplied.
logb xy = y × logb x
EX: log(26) = 6 × log(2) = 1.806
It is also possible to change the base of the logarithm using the following rule.
logb(x) = logk(x) / logk(b)
EX: log10(x) = log2(x) / log2(10)
To switch the base and argument, use the following rule.
logb(c) = 1 / logc(b)
EX: log5(2) = 1 / log2(5)
Other common logarithms to take note of include:
logb(1) = 0
logb(b) = 1
logb(0) = undefined
limx→0 logb(x) = −∞
ln(ex) = x
Frequently Asked Questions
How do you solve a logarithm equation for the unknown?
In logb(x) = y, any one value can be found from the other two: y = logb(x) (the log itself), x = by (rewrite the equation in power form), and b = x1/y (take the y-th root of x). Enter any two values and this calculator solves for the third.
What is the difference between log, ln, and log2?
They are the same operation with different bases: log usually means base 10 (common in science and engineering), ln means base e (the natural logarithm, common in math and physics), and log2 means base 2 (common in computer science).
Why can't the base of a logarithm be 1 or negative?
A base of 1 never changes as it is raised to powers (1 to any power is 1), so log base 1 is undefined except trivially. Negative bases produce values that jump between positive and negative and are undefined for most exponents, so logarithms require a positive base other than 1.
What is the log of a negative number or zero?
For real-number logarithms, the argument x must be positive: no power of a positive base produces zero or a negative number, so log of 0 or of a negative value is undefined. As x approaches 0 from above, logb(x) falls toward negative infinity (for b > 1).