Logarithm is inverse operation of an exponent. We know that 10³ =1000, this operation in logarithm can be written as log₁₀ 1000 = 3. That means 3 is the logarithm of 1000 to base 10. 

In general if \(a\) be a real number greater than 0, \(a \ne 1\) and \({a^x} = m\). Then \(x\) is called the logarithm of \(m\) to the base \(a\) and is written as \({\log _a}m\), and conversely, 

if  \({\log _a}m = x\), then \({a^x} = m\)

Note: Logarithm to a negative base and logarithm of negative quantity is not defined. 

\({\log _a}m\) is defined when \(m > 0\) and \(a > 0,\,\,a \ne 1\).

Since \({a^0} = 1\) \( \Rightarrow {\log _a}1 = 0\)

Also \({\log _a}a = 1\,,\,\,{\rm{for}}\,\,{\rm{all}}\;\,a > \,0,\,\,a \ne \,1\)

Graph of a log function:

The graph of \(y = {\log _a}x\) is an increasing function when \(a > 1\), that means when \(x\) increases, \(y\) also increases.  When \(0 < a < 1\), the graph of \(y = {\log _a}x\) decreases when \(x\) increases. Both the functions intersect at the point (1, 0).