Exponential Equations

In an exponential equation the unknown is in the exponent, for example 2ˣ = 32. The easiest way is to write both sides as powers with the same base: 32 = 2⁵. If two powers with the same base are equal, their exponents are equal too, so x = 5.

What does this generator do?

On the easy level both sides of the equation already have the same base, for example 3^(x − 1) = 27. On the medium level you rewrite both sides as powers of the same number, for example 4ˣ = 8 or 9^(x + 1) = 27ˣ, and the solution may be a fraction. On the hard level you solve equations using the substitution t = 2ˣ or by factoring out a common power.

How to use it

1) Work out which number both sides are powers of: 4 = 2², 8 = 2³, ¼ = 2⁻². 2) Rewrite the equation in the form 2^(…) = 2^(…). 3) Set the exponents equal and solve the linear equation. 4) For an equation like 4ˣ − 6 · 2ˣ + 8 = 0, let t = 2ˣ, solve the quadratic equation and go back to x. Discard a negative t — a power is always positive.