Can anyone solve this equation?

Question

log (base 5) 2x/log (base 5) 10 = ln 2e^2

pls give your answer to 3 significant figures and include the steps as well

Pls help...6 lives depend on it!!!

Answer 1

ln 2e^2 = ln2 + lne^2 = ln2 + 2

log(base 5)2x / log(base 5) 10 = log (base 10) 2x

[ Because log (base a) b = log b/log a ]

Now,

log 2x = 2 + ln2 // 10^both sides

2x = 10^(2 + ln2) = 100 * 10^ln 2

x=50*10^ln 2 = 5 * 10*10^ln2 = 5*10^(1 + ln2)

and I will use my calculator ...

x = 246.670...

Answer 2

..log (base 5) 2x/log (base 5) 10

1)learn all the basic properties of logarithms!!!!!!
2)log (base 5) 2x/log (base 5) 10
=log(base 10)2x
......the denominator and the numerator both have the same base so one of the thms allow us to get to this result!(try showing that log(base k)t/log(basek)m=log(basem)t as an exercise)

ln 2e^2=ln2+lne^2=ln2+2lne=ln2+2 ....since log(base k)k=k (again try proving that on your own or check your texbook/notes) remember lnk=log(base e)k where e is euler"s number(2.71828 18284 59045 23536.....)

so,log(base 10)2x =ln2+2
2x=10^(ln2+2).... ....again check that if log(base m)r=j then r=m^j

so x=(10^(ln2+2))/2