Logarithm as Inverse of Exponential
Main Concept

Given $b\>0$ and $xgt;0$, with $b\ne 1$, the logarithm base $b$ of $x$, written ${\mathrm{log}}_{b}\left(x\right)$ is the exponent to which $b$ needs to be raised to obtain $x$. That is, ${\mathrm{log}}_{b}\left(x\right)\=y$ means exactly that ${b}^{y}\=x$. Thus, the functions ${\mathrm{log}}_{b}\left(x\right)$ and ${b}^{x}$ are inverses of each other. The domain of the logarithm base $b$ is all positive numbers. The range of the logarithm base $b$ is all real numbers.

General Logarithms


Recall that the domain and range of an invertible function are just the range and domain of its inverse. Thus, the domain of the logarithm base $b$ function is the range of the ${b}^{x}$ function (all positive numbers) and the range of the logarithm base $b$ function is the domain of the ${b}^{x}$ function (all numbers).
Examples:
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${\mathrm{log}}_{2}\left(8\right)\=3$ since ${2}^{3}\=8$

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${\mathrm{log}}_{10}\left(.001\right)\=3$ since ${10}^{3}\=.001$

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${4}^{{\mathrm{log}}_{4}\left(7\right)}\=7\={\mathrm{log}}_{4}\left({4}^{7}\right)$ since the logarithmic function ${\mathrm{log}}_{4}\left(x\right)$ and the exponential function ${4}^{x}$ are inverses of each other.

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${\mathrm{log}}_{b}\left(1\right)\=0$ for any base $b$, since ${b}^{0}\=1$ for all $b\>0$.



The Natural Logarithm Function


One exponential function is so important in mathematics that it is distinguished by calling it the exponential function. This exponential function is written as ${\ⅇ}^{x}$ or, particularly when the expression in the exponent is complicated, $\mathrm{exp}\left(x\right)$. The inverse of this function is just as important in mathematics.
The Natural Logarithm Function

The natural logarithm function is the inverse of the exponential function, ${\ⅇ}^{x}$, where $\ⅇequals;2.718281828..period;$ . This function is so important in mathematics, science, and engineering that it is given the name "ln": $\mathrm{ln}\left(x\right)\={\mathrm{log}}_{\ⅇ}\left(x\right)$. Reading out loud, it is pronounced "lawn of x" or often just "lawn x".



The graph of the natural logarithm function can be obtained from that of the exponential function by reflection across the line $y\=x$:





Exploring the function log_{b}(a) with base greater than 1 and between 0 and 1


Use the sliders below the graphs to change the values of $b$, the base of the logarithmic function $yequals;{\mathrm{log}}_{b}\left(x\right)$ and its corresponding exponential function $y\={b}^{x}$. For the graph on the left, the base is a number greater than 1. For the graph on the right, the base is a number between 0 and 1. Note that there is no logarithmic function with base $b\=1$. Do you see why not?

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