Defining Logarithms

Logarithms Determine an Unknown Exponent

A logarithm reverses exponentiation. If $a^c=b$ , then $\log_a b$ gives the exponent $c$ .

\log_a b=c\quad\Longleftrightarrow\quad a^c=b

For real logarithms, the conditions are:

a>0\qquad a\neq1\qquad b>0

$a$ is the base
$b$ is the argument
$c$ is the exponent returned by the logarithm

The conditions match the behavior of the exponential function. A valid exponential base is positive and not equal to one, and $a^c$ is always positive, so a real logarithm cannot have a nonpositive argument.

Converting Between the Two Forms

Each row states the same relationship twice:

Exponential form	Logarithmic form
$2^5=32$	$\log_2 32=5$
$3^2=9$	$\log_3 9=2$
$5^{-2}=\frac1{25}$	$\log_5\frac1{25}=-2$
$7^0=1$	$\log_7 1=0$

When converting, keep the base unchanged. The exponential output becomes the logarithm's argument, and the exponent becomes the logarithm's value.

Common and Natural Logarithms

A logarithm with base $10$ is called the common logarithm. Its base is often omitted:

\log_{10}x=\log x

A logarithm with base $e$ is called the natural logarithm and is written $\ln x$ . Both notations are special cases of $\log_a x$ .

Solving for an Unknown Exponent

Suppose an idealized culture starts with $2{,}000$ bacteria and doubles every hour:

P(x)=2{,}000\cdot2^x

To find when the continuous model reaches $100{,}000$ , set the output equal to the target:

100{,}000=2{,}000\cdot2^x

Using a calculator or change of base:

x=\frac{\ln50}{\ln2}\approx5.643856

The continuous curve crosses $100{,}000$ at about $5.64\text{ hours}$ . If the culture is counted only after completed hourly divisions, the interpretation changes:

P(5)=2{,}000\cdot2^5=64{,}000

So the first completed hourly interval at or above the target is $6\text{ hours}$ . The logarithm solves the equation. The context determines how that real-valued answer should be interpreted.