Question 35303
if you are just learning about logs etc, this is a nasty question to give you.


Take logs to base5: i shall just write the word "log" without a subscript.


{{{ log(5^(sin2x)) = log(3.307) }}}
{{{ (sin2x)log(5) = log(3.307) }}}


Now, the reason for taking logs to base 5 is that log to base 5 of (5) is 1. So:
{{{ sin2x = log(3.307) }}}


Now, what the hell is log-base5 of (3.307)...your calculator doesn't store base5 logs. It does store base10 though, so we are going to convert the base5 into a base10. Pretty straightforward really:


{{{ sin2x = log(3.307)/log(5) }}} --> here the 2 logs are now both base10. Work the right hand side out on your calculator to give:


sin2x = 0.743142 to 6dp


so, now need to solve this trig equation. What is the domain of x? Since sin is a repeating function, you will have an infinite number of solutions, as it stands.



I shall do the "obvious" one:
2x = sin^(-1)(0.743142)
2x = whatever
x = whatever/2


--> i do not have a calculator handy to find this angle for you.


jon.