write a log (8 x 4 square root of 5 divide by 81) in terms of log 2, log3 and log 5

Question

log( 8 timus 4 root 5 divide by 81) it must be in fraction the expressed in terms of log 2, log 3 and log 5

Answer 1

8=3log2; 4=2log2; sqrt(5)=(1/2)log5; 81=4log3; so the expression is:

3log2+2log2+(1/2)log5-4log3=5log2+(1/2)log5-4log3.

But 2*5=10 and if logs are to base 10, log2+log5=log10=1 so log2=1-log5. The expression becomes:

5(1-log5)+(1/2)log5-4log3=5+((1/2)-5)log5-4log3=5-4.5log5-4log3.

This final expression is in terms of the required logs as shown in the second line of this solution (highlighted). I've taken this further because, if logs are to base 10, there is a relation between log₁₀2 and log₁₀5. Hence the enhanced solution in red.

Comments

I've clarified my solution. The final answer (to write in terms of the logs) was given early in the solution. I've just taken it a bit further to illustrate the relationship between logs of different numbers, in particular logs to base 10.

If log implies a different or a general base then the highlighted (bold) solution applies.

My answers are to clarify and explain and perhaps encourage the student to look deeper than the bare question itself and stimulate an interest in the vast subject of mathematics, through which the students can tackle questions themselves with enthusiasm, without having to resort to websites like this just to get their homework done.