### Logarithms and Exponents

**exponent**is a number that indicates how many times a base is multiplied by itself and shown by a raised number, such as:

**logarithm**is the exponent $x$ to which a base $b$ is raised to produce a given number $y$, written as:

Exponential Form | Logarithmic Form |
---|---|

$y=b^x$ |
$x=\log_b{y}$ |

$8=2^x$ |
$x=\log_2{8}$ |

$8=2^3$ |
$3=\log_2{8}$ |

Some types of logarithms are used frequently. One is the common logarithm. A **common logarithm** is a logarithm with base 10, or the exponent to which 10 is raised to equal a given number; written as $\log{x}$, where $x$ is the given number. Notice that a common logarithm can be written without a subscript indicating the base. For example, $\log{100}$ is the same as $\log_{10}{100}$.

The irrational number $e$ is used frequently in logarithmic and exponential applications. Note that $e$ represents a constant and not a variable. The number $e$ is a nonterminating, nonrepeating decimal with a value of approximately 2.718281845.... A **natural logarithm** is a logarithm with base $e$, or the exponent to which $e$ must be raised to equal a given number. A natural logarithm is written as $\ln{x}$, where $x$ is the given number. So, $\ln{2}$ is the same as $\log_e{2}$.

Rewrite $\log{1\rm{,}000}$ in exponential form.

Since there is no base indicated, the expression is a common logarithm, meaning that the base is 10:*log*for the common logarithm and

*ln*for the natural logarithm. To evaluate $\log{900}$, enter log(900), which gives an approximate result of 2.95. To evaluate $\ln{900}$, enter ln(900), which gives an approximate answer of 6.80. Logarithms in other bases must be converted to either base 10 or base $e$ to be evaluated with a calculator.

### Inverses of Exponential Functions

An **input** is a value in the domain of a function. An **output** is a value in the range of a function. For an exponential function of the form $f(x)=b^x$, the inputs of the function are exponents and the outputs are powers of the base $b$.

An **inverse function** is the result of switching the inputs and outputs of a given function when the result is also a function. The composition of a function and its inverse is the identity function. If the inputs and outputs of the exponential $f(x)=b^x$ are switched to produce an inverse function, the inputs of the inverse function will be powers of the base $b$, and the outputs will be exponents. In other words, the rule for the inverse function will be a logarithm.

**logarithmic function**, where $b>0$ and $b\neq 1$, has the form:

Write the equation of the inverse exponential function.

The inverse of the given logarithmic function is the exponential function:

Complete a table of values for the inverse exponential function.

$x$ |
$f(x)=2^x$ |
---|---|

$-2$ |
$0.25$ |

$-1$ |
$0.5$ |

$0$ |
$1$ |

$1$ |
$2$ |

$2$ |
$4$ |

Complete a table of values for the logarithmic function by switching the input and output values for the inverse exponential function.

$x$ |
$f(x)=\log_2{x}$ |
---|---|

$0.25$ |
$-2$ |

$0.5$ |
$-1$ |

$1$ |
$0$ |

$2$ |
$1$ |

$4$ |
$2$ |

Plot ordered pairs from the table for the inverse exponential function:

Then, connect them with a smooth curve.

Plot ordered pairs from the table for the logarithmic function:

Then, connect them with a smooth curve.

The curves are reflections of each other across the diagonal line $f(x)=x$.