Log to base 4 of (x-3)^3 or log to base 10 of 4(x-3)^3 or natural log of 4(x-3)^3? I'll answer all three.
Log to base 4 of (x-3)^3=4
The log of the cube of a quantity is 3 times the log of the quantity, so 3log(x-3)=4 and log(x-3)=4/3. To solve this raise both sides as powers of 4: x-3=4^4/3, so x=3+4^4/3=3+6.35=9.35. To find 4^4/3 I took 4(ln4)/3=1.8484 and then calculated e^1.8484=6.35. I could also have calculated using log to base 10 instead of e.
Log to base 10 of 4(x-3)^3=4
Log4(x-3)^3=log4+3log(x-3)=4. So 3log(x-3)=4-log4=3.39794 and log(x-3)=3.39794/3=1.13265.
This time we raise both sides as powers of 10: (x-3)=10^1.13265=13.572, so x=16.572.
Natural log of 4(x-3)^3=4
Ln4(x-3)^3=ln4+3ln(x-3)=4. So 3ln(x-3)=4-ln4=2.6137 and log(x-3)=2.6137/3=0.8712.
Raise both sides as powers of e: (x-3)=e^0.8712=2.3899, so x=5.3899.