Question 823335
{{{log(0.25, (sqrt(2))) - log(sqrt(2), (0.25))}}}
The key to the first problem is to recognize that 0.25 and sqrt(2) are both powers of 2:
{{{0.25 = 1/4 = 2^-2}}}
{{{sqrt(2) = 2^(1/2)}}}
So by using the change of base formula to change these two logs into expressions of base 2 logs, we will get something that should simplify easily. The change of base formula:
{{{log(a, (p)) = log(b, (p))/log(b, (a))}}}
Using this on both logs we get:
{{{log(2, (sqrt(2)))/log(2, (0.25)) - log(2, (0.25))/log(2, (sqrt(2)))}}}
Since we already know what powers of 2 produce 0.25 and sqrt(2), we can replace all the logs:
{{{(1/2)/(-2) - (-2)/(1/2)}}}
Now we simplify:
{{{(1/2)(1/(-2)) - ((-2)/1)(2/1)}}}
Multiplying:
{{{(-1/4) - (-4)}}}
which simplifies as follows:
{{{(-1/4) + 4}}}
{{{(-1/4) + 16/4}}}
{{{15/4}}}
{{{3&3/4}}} or 3.75