%path = "maths/functions/log" %kind = kinda["texts"] %level = 10

Logarithm

The power operation generates a result from the basis and the exponent. So from the result there are two ways back: either to the basis or to the exponent.

From the calculation rules of exponents with same basis, e.g. \(2^32^2=2^{3+2}\) and \(\frac{2^3}{2^2}=2^{3-2}\) follow the logarithm rules that make plus out of multiply and minus out of divide.

The repetition of multiplication (power) becomes repetition of addition (multiplication).

\[\begin{split}\begin{matrix} \log ab &= \log a + \log b \\ \log \frac{a}{b} &= \log a - \log b \\ \log b^c &= c\log b \end{matrix}\end{split}\]

From the last rule it follows how to calculate any logarithm with just one logarithm.

\[\begin{split}b^x &= d \\ x &= \frac{\log d}{\log b}\end{split}\]

An exponential equation, i.e. an equation that has the unknown in the exponent, is solved best by first trying to bring it into the form \(b^x = d\) and then apply the logarithm on both sides.

The logarithm always refers to a basis. If the basis is not specified, then \(\log\) is either with Basis 10 or with basis e=2.71828182846… (Euler number)

It is

\[\begin{split}\log_{10} 10 = \log 10 = \text{lg} 10 = 1\\ \log_e e = \ln e = 1\\ \log_2 2 = \text{lb} 2 = 1\\\end{split}\]