Log calculator
This log calculator computes the result of raising a given base to a specified power. It offers flexibility by allowing you to either input your own base and power values or select “e” as the base with the option to customize the default power value. You can easily switch between using a custom base and power or “e” as the base, and this log calculator provides real-time updates of the result as the input values change.
| log | |
| Base | Power | Result |
|---|---|---|
| 5 | - | - |
Related calculators:
What is a log?
In mathematics, a logarithm, often abbreviated as “log,” is a mathematical function that expresses the relationship between exponential growth and repeated multiplication. The logarithm of a number to a certain base is the exponent to which the base must be raised to obtain that number. The most common bases used in mathematics are 10 (common logarithm) and the number “e” (natural logarithm).
The general formula for the logarithm of a number “x” to the base “b” is expressed as:
\log_b(x)
where:
- x is the number for which you want to find the logarithm.
- b is the base of the logarithm.
There are two widely used logarithmic bases:
- Common Logarithm (base 10): log_{10}(x) is usually denoted as log(x) without specifying the base. This is known as the common logarithm.
- Natural Logarithm (base “e”): log_e(x) is denoted as ln(x) , where “ln” stands for the natural logarithm, and the base “e” is approximately equal to 2.71828.
Logs are widely used in fields like engineering, mathematics, statistics, and science. They simplify complex calculations, transform data for analysis, model exponential growth and decay, and appear in algorithm analysis, signal processing, and more. In finance, logs calculate returns, while in chemistry, they describe reaction kinetics. Logs play crucial roles in information theory, environmental science, and even music theory, showcasing their versatility across various disciplines.
Logarithmic rules
There are several logarithm rules that can be useful in simplifying expressions and solving equations involving logarithms.
Product rule:
This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors when both factors are taken with the same base b.
\log_b(xy) = \log_b(x) + \log_b(y)
Quotient rule:
This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator when both numerator and denominator are taken with the same base b.
\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)Power rule:
This rule states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the base when both the number and the base are taken with the same base b.
\log_b(x^n) = n \times \log_b(x)
Change of base formula:
If you have a logarithm with a base that you can’t easily work with, you can use the change of base formula to express it in terms of a different base. The formula is as follows:
\log_b(x) = \frac{\log_c(x)}{\log_c(b)}where:
- \log_b(x) is the original logarithm.
- \log_c(x) and \log_c(b) are logarithms with the new base “c,” which is typically chosen to be a base for which you have a calculator (e.g., base 10 or base “e”).
Logarithm of 1:
The logarithm of 1 to any base is always 0:
\log_b(1) = 0 \text{ for any positive base } b Logarithm of the base:
The logarithm of a number to its own base is always 1:
\log_b(b) = 1 \text{ for any positive base } b Negative logarithm:
If you have a negative number inside the logarithm, it’s undefined in the real number system:
\log_b(x) \text{ is undefined for } x \leq 0 Logarithm of a fraction:
You can express the logarithm of a fraction as the difference of the logarithms of the numerator and denominator:
\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \\ \text{ for } x > 0 \text{ and } y > 0