Properties of Logarithms - Maple Programming Help

Home : Support : Online Help : Math Apps : Functions and Relations : Logarithmic Functions : MathApps/PropertiesOfLogarithms

Properties of Logarithms

Properties of Logarithmic Functions

Let $b>0$, $b\ne 1$, let $x$ and $y$ be positive numbers, and let $r$ be any real number. Then the following properties hold:

 1 The range of ${\mathrm{log}}_{b}\left(x\right)$ is all real numbers.
 2 The domain of ${\mathrm{log}}_{b}\left(x\right)$ is all positive real numbers.
 3 For $b>1$, ${\mathrm{log}}_{b}\left(x\right)>0$ for $x>1$ and ${\mathrm{log}}_{b}\left(x\right)<0$ for $0; for $0 the inequalities reverse.
 4 ${\mathrm{log}}_{b}\left(1\right)=0$
 5 ${\mathrm{log}}_{b}\left(x\cdot y\right)={\mathrm{log}}_{b}\left(x\right)+{\mathrm{log}}_{b}\left(y\right)$
 6 ${\mathrm{log}}_{b}\left(\frac{x}{y}\right)={\mathrm{log}}_{b}\left(x\right)-{\mathrm{log}}_{b}\left(y\right)$
 7 ${\mathrm{log}}_{b}\left(\frac{1}{x}\right)=-{\mathrm{log}}_{b}\left(x\right)$
 8
 9 If $x>y$ and $b>1$ then ${\mathrm{log}}_{b}\left(x\right)>{\mathrm{log}}_{b}\left(y\right)$. If $x>y$ and $0 then ${\mathrm{log}}_{b}\left(x\right)<{\mathrm{log}}_{b}\left(y\right)$. That is, ${\mathrm{log}}_{b}\left(x\right)$ is an increasing function if $b>1$ and a decreasing function if $0.
 10 $x={\mathrm{log}}_{b}\left(y\right)$ exactly when $y={b}^{x}$. That is, the logarithmic and exponential functions with the same base are inverses of each other. In particular, ${\mathrm{log}}_{b}\left({b}^{x}\right)=x={b}^{{\mathrm{log}}_{b}\left(x\right)}$.

Using the properties of logarithms

The calculator shown here is missing a few keys (no multiplication or division keys). Nonetheless, it is still possible to perform any arithmetic calculation involving only +, -, ×, or ÷ operations. This is because the calculator has "10 to the power of" and "logarithm base 10" keys.

Try it out. Can you compute these values?

 • 4/3
 • $93123×23485$
 • $\frac{\left(5217.308+234.33×941.226\right)}{177.332}$

The "log" button represents the base 10 logarithmic function. The calculator displays answers to 2 decimal places.



 More MathApps