In its simplest form, a logarithm is an exponent. Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.

Logarithms have the following structure: $log{_b}(x)=c$ where $b$ is known as the base, $c$ is the exponent to which the base is raised to afford $x$. The base $b>0$.

Note that $​​log{_b}x=c$ is not defined for $c<0$. This is because the base $b$ is positive and raising a positive number to any power will yield a non-negative number.

Commonly Used Bases

A logarithm with a base of $10$ is called a common logarithm and is denoted simply as $logx$. The common log is used often in science and engineering.

A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$. The irrational number  $e\approx 2.718$ arises naturally in financial mathematics, in computations having to do with compound interest and annuities.

A logarithm with a base of $2$ is called a binary logarithm. While it has no special notation, it is often used in computer science.

The Exponential and Logarithmic Forms of an Equation

Logarithmic equations can be written as exponential equations and vice versa. The logarithmic equation $log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.

As an example, the logarithmic equation $log{_2}16=4$ corresponds to the exponential equation $2^4=16$.

Example 1: Solve for $x$ in the equation $log{_3}(243)=x$. 

Here we are looking for the exponent to which $3$ is raised to yield $243$. 

It might be more familiar if we convert the equation to exponential form giving us:

$3^x=243 \\ 3^5 =243$ 

Thus, $log{_3}(243)=5$.

The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with $243$, if we take its logarithm with base $3$, then raise $3$ to the logarithm, we will once again arrive at $243$.

Trivial Logarithmic Identities

$log_{b}1=0$ as  $b^0=1$ for $b\neq 0$. Note that $0^0\neq 1$. Rather, $0^0$ is called an indeterminate form.

$log{_b}b=1$ as $b^1=b$

$log{_b}0=undefined$, as there is no number x such that $b^x=0$

The following two logarithmic identities can be verified by converting the logarithmic equation into an exponential equation as follows:

$b^{log_{b}(x)}=x$ 

Converting this to a logarithmic equation yields: $log_{b}(x)=log_{b}(x)$

Converting $log_{b}(b^x)=x$ to an exponential equation yields $b^x=b^x$

Applications of Logarithms

Historically, logarithms were invented by John Napier as a way of doing lengthy arithmetic calculations prior to the invention of the modern day calculator. 

More recently, logarithms are most commonly used to simplify complex calculations that involve high-level exponents. In chemistry, for example, pH and pKa are used to simplify concentrations and dissociation constants, respectively, of high exponential value. The purpose is to bring wide-ranging values into a more manageable scope. A dissociation constant may be smaller than $10^{10}$, or higher than $10^{-50}$. Taking the logarithm of each brings the values into a more comprehensible scope ($10$ to $-50$) .