We're given the following function:
[tex]f(x)=log(.75^x)=log[ (\frac{3}{4}) ^{x}]=log( \frac{3^x}{4^x}) [/tex]
In order to see if the function is decreasing we'll take its derivative. If [tex] \frac{d}{dx}f(x)\ \textgreater \ 1 [/tex] the function is increasing, if [tex]\frac{d}{dx}f(x)\ \ \textless \ \ 1 [/tex] the function is decreasing.
We take the derivate:
[tex] \frac{d}{dx}[log( \frac{3^x}{4^x})]= \frac{4^x}{3^x}[ \frac{d}{dx} (\frac{3^x}{4^x})]=\frac{4^x}{3^x} \frac{4^x[ \frac{d}{dx} (3^x)]-3^x[ \frac{d}{dx} (4^x)]}{ 4^2^x}=[/tex]
[tex]\frac{4^x}{3^x} \frac{4^x[3^xlog(3)]-3^x[4^x[4^xlog(4)]}{ 4^2^x}=log(3)-log(4)\ \textless \ 0[/tex]
Which implies the function is decreasing.
Another way to answer the problem (although less insightful) you can take any two real numbers [tex]k[/tex] and [tex]q[/tex] such that [tex]k\ \textgreater \ q[/tex], then if [tex]f(k)-f(q)\ \textgreater \ 0[/tex] the function is increasing and if [tex]f(k)-f(q)\ \textless \ 0[/tex] the function is decreasing. You can verify the function is decreasing with any two numbers in the function's domain.
