When solving the problem 7^x+2=7(3x)-8, your friend does the following:

7(x)+2=7(3x)-8
7x+2=21x-8
10=14x
x=5/7
was this correct?

No, the calculation was not correct. The mistake occurred when your friend distributed the 7 on the right side of the equation. It should be 7 * 3x = 21x, not 7 * x = 7x.