Exponential equations and logarithms

What an exponential equation is

• An exponential equation has the unknown in the index (the power), like $2^x = 20$.
• There are two ways to solve one:
  • Equate the indices — when both sides can be written with the same base.
  • Take logarithms — when the bases cannot be matched.

Solving by equating indices

• If both sides can be written as powers of the same base, then the indices must be equal.
• Rewrite any numbers as powers of that base first — e.g. $81 = 3^4$ and $32 = 2^5$.
• $3^{x+1} = 81 = 3^4$, so $x + 1 = 4$ and $x = 3$.

What a logarithm is

• A logarithm is the inverse of a power — it answers the question "what power?".
• $\log_a b = c$ means exactly $a^c = b$. So $\log_2 8 = 3$ because $2^3 = 8$.
• Because logs undo powers, they let you get at an unknown that is stuck in an index.
• The log laws mirror the index laws:

$\log(xy) = \log x + \log y \qquad \log\!\left(\tfrac{x}{y}\right) = \log x - \log y \qquad \log(x^n) = n\log x$

• The most useful is the power law $\log(x^n) = n\log x$ — it lets you bring a power down to the front as a multiplier.

Solving exponential equations with logs

• When the bases cannot be matched, take the log of both sides.
• Use the power law to bring the unknown index down to the front.
• Divide to isolate the unknown.