Logarithms as Inverse Exponentials

Throughout suppose that $a>1$. The function $y=\log_a(x)$ is the inverse of the function $y=a^x$. In other words,

whenever these make sense.

• Since $10^3=1000$, $\log_{10}(1000)=3$.


• Since $2^{-3}=1/8$, $\log_2(1/8)=-3$.


• $10^{\log_{10}(17)}=17$ and $2^{\log_2(.038)}=.038$


• Since $a^0=1$, no matter what $a$ is, $\log_a(1)=0$.  In general $\log_a(x)$ will be positive for $x>1$ and negative for $x<1$.


• The domain of $a^x$ is $(-\infty, \infty)$, so the range of $\log_a(x)$ is $(-\infty,\infty)$.


• The range of $a^x$ is $(0,\infty)$, so the domain of $\log_a(x)$ is $(0,\infty)$. We cannot take the log of zero or the log of a negative number.
  

Laws of Logarithms

The laws of logs follow from the law of exponents.  You should know these.  Provided that $x>0$ and $y>0$ we have:

Graphs

Below are the graphs of the log functions when $a>1$ or $0 < a<1$.