calculus logarithmic differentiation question help

Question

Use logarithmic differentiation to find the derivative of 𝑦=(ln(7𝑥))^(ln(2𝑥)):

Answer 1

y=(ln(7x))^ln(2x)

Take log of both sides

ln(y)=ln((ln(7x))^ln(2x))

ln(y)=ln(2x) ln(ln(7x)) log identity ln(a^b) = bln(a)

Take derivative

y'/y = (1/(2x))(2)ln(ln(7x)) + ln(2x)(1/ln(7x))(1/(7x))(7)

y' = y[ln(ln(2x))/x +ln(2x)(1/ln(7x))(1/x)]

where y = (ln(7x))^ln(2x)