Rewriting the equation so that each side of the equation is a power of the same base. This cannot always be done but it is easier than the alternative so it is worth looking for this.
Isolate the base and its exponent ans use logarithms.
128 is not a integral power of 4 so we cannot easily write both sides as powers of 4. But 4 and 128 are both powers of 2! So we will be able to rewrite both sides of the equation as powers of 2:
On the left side we're raising a power to a power. The rule for exponents for this is to multiply the exponents:
We now have an equation that says that two powers of 2 are equal. The only way this can be true is if the exponents are equal, too. So:
2x = 7
Now that x is no longer in an exponent we can solve for it. Dividing by 2 we get:
Note: If you use logarithms well you should get 7/2 that way, too. However, it's more difficult and still requires that you figure out the both 4 and 128 are powers of 2.
(