# Index laws with different bases of a relationship

### Indices_and_logarithms

Log base 2 is defined so that In other words, the logarithm gives the exponent as the output if you give it the exponentiation result as the input. To get . we determine that a relationship between the natural log and the exponential function is. Once index notation is introduced the index laws arise naturally when simplifying numerical and algebraic expressions. 1: To multiply powers with the same base, add the indices. .. The relationship connecting logarithms and powers is. The fundamental index laws are given by \$latex \begin{aligned} x^a x^b &= x^{a + b}\quad\quad &(1)\\(x^a)^b &= x^{ab}\quad Case 1: Positive bases It is also unbounded below due to the relationship \log(1/x) = -\log(x).