There are two bases for logarithms that are so commonly used they receive a special designation: Base 10 and the natural base \(e\):

\[ \solve{\log_{{10}}(A) &=& \log(A)\\ \log_{{e}}(A) &=& \ln(A)} \]

The base 10 logarithm is implied whenever we write just \(\log\) whereas the \(\ln\) actually stands for the French "logarithm natural" (aka, logarithm with the natural base, usually called the natural log in English).

Given any standard exponential base, \(y=B^x\), the domain of this exponential function is all real numbers, a vertical intercept at \((0,1)\), the range is \((0,\infty)\), and the graph has a horizontal asymptote at \(y=0\). The corresponding logarithmic equation \(y=\log_B(x)\) swaps the domain and range so it has a domain of \((0,\infty)\) with a vertical asymptote at \(x=0\), a horizontal intercept at \((1,0)\), and has a range of all real numbers.