Powers of powers and exponential equations

In an exponential equation the unknown is in the exponent. The key idea: for a positive base other than 1, different exponents give different results (2¹ = 2, 2² = 4, 2³ = 8 …), so if aˣ = aʸ then x = y. We need a > 0 and a ≠ 1 because 1ˣ = 1 for every x, which tells us nothing about x, and a negative base would change sign from one power to the next. The plan: (1) write each number as a power of a small prime, for example 81 = 3⁴, 125 = 5³, 32 = 2⁵, 1 ÷ 9 = 3⁻²; (2) use the laws to make both sides one power of the same base; (3) equate the exponents and solve the ordinary equation; (4) check by substitution. Equations such as 2ˣ = 0 or 3ˣ = −9 have no solution, because a positive base never gives zero or a negative number.