Mathematical tables: logarithm operations
Table shows common properties and formulas related to logarithms (logarithm function).

# Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

# Logarithms: general formulas#

 Name Formula Logarithm of one Show source$\log_{a} 1 = 0$ Logarithm of the product Show source$\log_{a} \left(x \cdot y \right) = \log_{a} x + \log_{a} y$ Logarithm of the quotient Show source$\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$ Sum of logarithms with common base Show source$\log_{a} x + \log_{a} y = \log_{a} \left(x \cdot y \right)$ Difference of logarithms with common base Show source$\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}$ Logarithm from power Show source$\log_{a} x^n = n \log_{a} x$ Logarithm from n-th root Show source$\log_{a} \left( \sqrt[n]{x} \right) = \frac{\log_{a} x}{n}$ Logarithm base conversion Show source$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$

# Decimal logarithm #

 Name Formula Decimal logarithm from 1 Show source$\log_{10} 10 = 0$ Decimal logarithm from 10 Show source$\log_{10} 10 = 1$ Decimal logarithm from power of 10 Show source$\log_{10} 10^n = n$ Decimal logarithm to natural conversion Show source$\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x$ Natural logarithm to decimal conversion Show source$\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x$

# Natural logarithm #

 Name Formula Natural logarithm from 1 Show source$\ln e = 0$ Natural logarithm from e number Show source$\ln e = 1$ Natural logarithm from power of e number Show source$\ln e^n = n$ Decimal logarithm to natural conversion Show source$\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x$ Natural logarithm to decimal conversion Show source$\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x$

# Some facts#

• Calculation of logarithm is an operation opposite to exponentiation.
• To calculate the logarithm of base a of the number x we ask the question: "to what power we must raise the base to get x".
• ⓘ Example: The decimal logarithm (of the basis 10) of the number one hundred (100) is two (2), because to obtain this number (100), we would have to raise the base of logarithm (10) to the second power. Formally, we can write this down in the following way:
$\log_{10}100 = 2 \Leftrightarrow 10^2 = 100$
• Formally the logarithm is a two arguments operation, where the first argument is the logarithm base, and the second one is the number that we logarithm. In general, we can write:
$\log_{a} x = b \Leftrightarrow a^b = x$
where:
• a - the logarithm base,
• x - the number, which we logarithm,
• b - the result of logarithm operation
• The logarithm is usually denoted by the symbol log or lg.
• Sometimes to simplify the expression we omit the base of the logarithm. Then, by default, we assume that the basis is the number ten (10).
• A special type of logarithm is a natural logarithm. The basis of natural logarithm is so-called number e (sometimes also called the Euler's number):
$e = \sum\limits_{n = 0}^{\infty} \dfrac{1}{n!} = \dfrac{1}{1} + \dfrac{1}{1} + \dfrac{1}{1\cdot 2} + \dfrac{1}{1\cdot 2\cdot 3} + \cdots$
• In general, the logarithm is a inverse function for the power function $a^x$. Thus, the natural logarithm is a inverse to the exponential function $e^x$.
• The practical aspect of logarithm is that it allows to replace multiplication with addition, which often simplifies calculations:
$\log_a(x \cdot y) = \log_a(x) + \log_a(y)$

• Logarithm is often used by engineers and in natural science. This is because expressions that have an inherently exponential character (or more generally: power character), become linear after logarithm was applied. For this reason, many laws, e.g. in physics or chemistry have logarithm form.

# Tags and links to this website#

Tags:
Tags to Polish version: