Generally addition is easier than multiplication. In calculus the logarithm and exponent are important tools because they can translate problems about multiplication to problems about addition. They are functions such that \[ a^{x+y}=a^x\cdot a^y\text{ and }\log_a(xy)=\log_a(x)+\log_a(y), \] and the 17th century mathematician Napier used them to simplify complicated computations. For example, if he wanted to compute $145940^3$, he would look up the table of logarithms to see \[ \log_{10}(1.4594)=0.164\text{ so }\log_{10}(145940)=5.164. \] He would then use properties of logarithms to see \[ \log_{10}(145940^3)=5.164\cdot3=15.492, \] then look back at his table of logarithms to see $10^{0.492}=3.104$, so \[ 145940^3=3.104\times10^{15}, \] approximately.
Moreover, we could have chosen any base instead of $10$. For example logarithms and exponentials with base $2$ can be translated to logarithms and exponentials with base $10$ since \[ 2^x=10^{\log_{10}(2)x}\text{ and }\log_2(x)=\log_2(10)\cdot\log_{10}(x). \]
In number theory we are interested in working modulo a prime number $p$. There are crucial differences from exponentials and logarithms in calculus:
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