Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4* General Logarithmic and Exponential Functions - 6.4* Exercise: 68

Answer

$(ab)^{x}=a^{x}.a^{y}$ This is known as fourth Law of Exponents; where $x$ and $y$ are real numbers.

Work Step by Step

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.
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