# Logarithm **An operation which returns the exponent to which a fixed base must be raised to produce a given number.** ![[Logarithm.svg|500]] *Graphs of logarithms with different bases; every graph intersects the point $(1, 0)$ and $(b, 1)$, where $b$ is the base; every graph also never intersects the vertical axis* > [!NOTE] Logarithm > The **logarithm** $\log_{b}x$ is the exponent to which the *base* $b$ of the logarithm must be raised to give the number $x$. > > Equivalently, the function $\log_{b}x$ is the *inverse* function of the [[exponentiation]] of $b$ to the power of $x$. That is, > $x=b^{y}\implies y=\log_{b}x$ > > The logarithm $\log_{b}x$ is referred to most commonly as "the log, base $b$, of $xquot;. The *[[natural logarithm]]* $\ln x$ is a named function which is the logarithm with [[e|Euler's number]] as the base. The *[[complex logarithm]]* is an extension of the natural logarithm to [[Complex Number|complex numbers]]. [[Logarithm#Logarithmic scale|Logarithmic scales]] are used in place of linear scales in graphs such that quantities for which there is a wide range can be displayed more compactly. ## Logarithm laws The following laws are only applicable for *positive real* $m$ and $n$ and *integer* $p>1$. - $\log_{b}(mn)=\log_{b}m+\log_{b}n$ - $\log_{b}\frac{m}{n}=\log_{b}m-\log_{b}n$ - $\log_{b}(m^{p})=p\log_{b}m$ - $\log_{b}\sqrt[p]{m}=\frac{1}{p}\log_{b}{m}$ ### Change of base law > [!NOTE] Change of base law > The *change of base law* is a formula for the logarithm in base $b$ of a number $x$ expressed in terms of the logarithm of an arbitrary base $k$. > $\log_{b}x=\frac{\log_{k}x}{\log_{k}b}$ ## Logarithmic scale A *logarithmic scale* is a non-linear scale in which every division of the scale corresponds to a multiple of a chosen base raised to a power. Thus, values which are an equal distance apart are different by an order of magnitude of the base. Logarithmic scale are used when quantities which are to be plotted vary significantly. A plot in which both axes of the plot is a logarithmic scale is known as a *log-log plot*. A plot in which only one of the axes is logarithmic is known as a *semi-log plot*. ![[LogarithmicScale.svg]] *A logarithmic scale* ![[LogarithmSemiLog.svg]] *A semi-log plot with a linear horizontal axis and a logarithmic vertical axis; the functions appear as though the logarithm of them are plotted instead*