(4x)^lg5 = (5x)^lg7...Solve for x?

Question

please help. thanks

Answer 1

Take logs of both sides: log(5)log(4x)=log(7)log(5x),

log(4x)/log(5x)=log(7)/log(5)=1.209062 approx.

Let k=log(7)/log(5) (for convenience).

(log(4)+log(x))/(log(5)+log(x))=k; let y=log(x):

log(4)+y=k(log(5)+y)=klog(5)+ky,

log(4)-klog(5)=y(k-1),

y=(log(4)-klog(5))/(k-1). All values on the RHS are known. Assume natural logs.

y=-2.676794 approx=log(x), x=e^y=0.0687833 approx.