Solving exponential equations with a common base

Same base, same exponent

An exponential function is one-to-one: each output comes from exactly one input. That gives the key property.

So if you can write both sides as powers of the same base, set the exponents equal.

When there's no solution

With a positive base, \(b^x\) is always positive. So \(3^x = -9\) and \(4^x = 0\) have no solution.

Quadratic in form

If an equation contains \(b^{2x}\) and \(b^x\), notice that \(b^{2x} = (b^x)^2\) and substitute \(u = b^x\).

When the bases don't match

An equation like \(3^x = 10\) can't be rewritten with a common base. Its solution is "the exponent that turns \(3\) into \(10\)", a logarithm: \(x = \log_3 10 \approx 2.096\). You'll learn how to find it in the next lessons.

Practice