#### Chapter 6 Elementary Functions

Section 6.4 Exponential and Logarithmic Functions

# 6.4.5 Logarithm Rules

For calculations involving logarithmic functions certain rules apply that can be derived form the exponent rules.
##### Info 6.4.11

The following rules are called logarithm rules:

$\begin{array}{cccc}\hfill \mathrm{log}\left(u·v\right)& \hfill =\hfill & \mathrm{log}\left(u\right)+\mathrm{log}\left(v\right)\hfill & \left(u,v>0\right) ,\hfill \\ \hfill \mathrm{log}\left(\frac{u}{v}\right)& \hfill =\hfill & \mathrm{log}\left(u\right)-\mathrm{log}\left(v\right)\hfill & \left(u,v>0\right) ,\hfill \\ \hfill \mathrm{log}\left({u}^{x}\right)& \hfill =\hfill & x·\mathrm{log}\left(u\right)\hfill & \left(u>0,x\in ℝ\right) .\hfill \end{array}$

These rules do not only apply to natural logarithmic functions but also to all other logarithmic functions. They can be used to transform a given expression in such a way that the power occurs only in the logarithmic terms.
##### Example 6.4.12
For example, the value $\text{ld}\left({4}^{5}\right)$ can be calculated applying the logarithm rules:

$\text{ld}\left({8}^{5}\right)\mathrm{ }=\mathrm{ }{\mathrm{log}}_{2}\left({8}^{5}\right)\mathrm{ }=\mathrm{ }5·{\mathrm{log}}_{2}\left(8\right)\mathrm{ }=\mathrm{ }5·{\mathrm{log}}_{2}\left({2}^{3}\right)\mathrm{ }=\mathrm{ }5·3\mathrm{ }=\mathrm{ }15 .$

Products in logarithmic functions can be split into sums outside the logarithmic functions:

$\text{lg}\left(100·\sqrt{10}·\frac{1}{10}\right)\mathrm{ }=\mathrm{ }\text{lg}\left(100\right)+\text{lg}\left(\sqrt{10}\right)-\text{lg}\left(10\right)\mathrm{ }=\mathrm{ }2+\frac{1}{2}-1\mathrm{ }=\mathrm{ }\frac{3}{2} .$

Importantly, the splitting rule $\mathrm{log}\left(u·v\right)=\mathrm{log}\left(u\right)+\mathrm{log}\left(v\right)$ transforms products into sums. The other way round is impossible for logarithmic functions: the logarithm of a sum cannot be transformed any further.