## Calculus 8th Edition

$(ab)^{x}=a^{x}.a^{y}$ This is known as fourth Law of Exponents; where $x$ and $y$ are real numbers.
We already know that $b^{x}=e^{xlnb}$ Now, $(ab)^{x}=e^{xln(ab)}$ This implies $(ab)^{x}=e^{x(lna+lnb)}$ Using formula, $e^{x+y}=e^{x}.e^{y}$ $(ab)^{x}=e^{x(lna+lnb)}=e^{xlna}.e^{xlnb}$ Hence, $(ab)^{x}=a^{x}.a^{y}$ This is known as fourth Law of Exponents; where $x$ and $y$ are real numbers.