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=ey=0.0687833 approx.