Take the log of both sides.

$\log \left(7\right)\ =\ \log \left(4.2\right)^x$

Use the Power Property of Logarithms to rewrite the left side of the equation, then divide both sides of the equation by log (4.2).

$\log\left(7\right)\ =\ x\cdot\log\left(4.2\right)$

Divide both sides by $3$.

$x^5 = 42$

Take the fifth root of both sides to remove the radical.
You may use your calculator.

$x\ =\ \sqrt[5]{42}$

Solve the equation to isolate $4^x$.

$4^x = 12$

Now follow the steps in part (a).

$x ≈ 1.792$