The Logarithmic Function: Let b be a positive real number. For each positive real number x , the logarithm with base b of x is the exponent to which b must be raised to obtain x . The logarithmic function with base b is the function from R + to R that takes each positive real number x to "log base b of x."

is equivalent to .

> plot([log[2](x), x^(1/2)],x=0..20, color=[green, cyan], title="Logarithmic Vs. Square Root",legend=["lg(x)","sqrt(x)"]);

> plot([2^x, exp(x)], x=0..2, color=[red,blue], style=[point,line], title="Exponential Functions",legend=["2^x", "e^x"]);

> plot([x^7, x], x=0..10, color = [cyan, yellow], style=[point,point],symbol=[circle, box], title="Power Functions", legend=["Degree 7", "Linear"]);

>

Which of the above functions grow the fastest? the slowest? Sort them according to their best big-oh appproximations.BOX, CROSS, CIRCLE, POINT, and DIAMOND

> plot([log[2](x),2^x, exp(x),x^7, x], x=0..1.5, color = [green,red, blue,cyan, yellow], style=[line,point,line,point,point],symbol=[point,diamond,point,circle, box], title="Power, Log, & Exponential Functions",legend=["lg(x)","2^x", "e^x", "x^7", "x"]);

> plot([log[2](x),2^x, exp(x),x^7, x], x=10..11.5, color = [green,red, blue,cyan, yellow], style=[line,point,line,point,point],symbol=[point,diamond,point,circle, box], title="Power, Log, & Exponential Functions",legend=["lg(x)","2^x", "e^x", "x^7", "x"]);plot([log[2](x),2^x, exp(x),x^7, x], x=40..51.5, color = [green,red, blue,cyan, yellow], style=[line,point,line,point,point],symbol=[point,diamond,point,circle, box], title="Power, Log, & Exponential Functions",legend=["lg(x)","2^x", "e^x", "x^7", "x"]);

Notice that from x= 10 to 11.5, the power function is much larger than any of the other functions including the exponential functions. The exponential and power functions cross at two locations: x = ~1.18 and x = ~21.465. For values of x larger than this second crossing, the exponential function clearly dominates.

The order of the functions from most efficient to least efficient are log n , sqrt( n ), n , n lg n , n ^7, 2^ n , e ^ n .