\magnification 2200
\vsize 10truein
\voffset -0.4truein
\nopagenumbers
\parskip 8pt
\parindent 0 pt

\def\u{\vskip 3.3in}
\def\v{\vskip 2.2in}
\def\w{\vskip 10pt}
\def\z{\vskip 1.2in}

$a^x  a^y = a^{x + y}$
\hfill 
${a^x \over a^y} = a^{x - y}$

$(a^x)^y = a^{xy}$
\hfill
$(ab)^x = a^x b^x$

Ex:  $2^8 5^6$ =

Suppose $f(x) = a^x$

\vskip -10pt

~~~~~~~~~~~~$y = a^x$

Find $f^{-1}$


~~~~~~Switch $x$ and $y$: $a^y = x$
 
~~~~~~$log_a x = y$ iff $a^y = x$

$f^{-1}(f(x)) =$

$f(f^{-1} (x)) =$

$log_a x + log_a y = log_a (xy)$ 

$log_a x - log_a y = log_a ({x \over y})$ 

$log_a x^r  = r log_a (x)$ 

$log_a a =$ \hfil $log_a 1 =$ \hfil $log_a 0 =$ \hfil 


Defn:  $ln(x) = log_e x$


$log_a x = { ln(x) \over ln(a)}$


Note:  $log_a x + log_a y \not= log_a (x + y) $



\end
