solving exponential equations

An exponential equation hides its unknown up in an exponent, like 2^x = 50 or 5 · 3^(t-1) = 90. The challenge is that ordinary algebra — adding, dividing, factoring — cannot reach a variable trapped in an exponent. You need a tool that can climb up there and bring it down: the logarithm.

There are two main strategies. If both sides can be written as powers of the same base, just set the exponents equal: from 2^x = 2^5 you immediately get x = 5. When the bases cannot be matched, take a logarithm of both sides and use the power law to drop the exponent into ordinary multiplication, then isolate the variable as usual.

Always finish by checking your answer, especially since some setups introduce extraneous solutions when you square or combine terms. And remember the domain limits of logarithms: the input must be positive. An exponential like b^x is always positive, so an equation like 2^x = -4 has no real solution, no matter how you manipulate it.