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Matching bases

Exponential functions are one-to-one, so if two sides of an equation have the same base, the exponents must be equal:

a^x = a^y \;\Longrightarrow\; x = y \qquad (a>0,\ a\neq1)

This means the first move on any exponential equation is trying to rewrite every term with a common base (e.g. rewriting 8 and 32 as powers of 2).

Log properties

\log_a(xy) = \log_a x + \log_a y, \qquad \log_a\!\left(\dfrac{x}{y}\right) = \log_a x - \log_a y, \qquad \log_a(x^y) = y\log_a x

The change-of-base formula lets you convert between any two bases (useful for combining logs with mismatched bases, or for calculator-free comparison):

\log_a x = \dfrac{\log_b x}{\log_b a}

A log equation always carries a hidden domain condition: the argument of a log must be positive. Solve the equation algebraically, then discard any solution that would make an argument zero or negative.

Worked example

Problem: Solve $\log_2(x) + \log_2(x-2) = 3$ .

Solution: Combine the logs: $\log_2\big(x(x-2)\big) = 3$ . Convert to exponential form:

x(x-2) = 2^3 = 8 \;\Longrightarrow\; x^2 - 2x - 8 = 0 \;\Longrightarrow\; (x-4)(x+2)=0

So $x=4$ or $x=-2$ . Check the domain: we need $x>0$ and $x-2>0$ , i.e. $x>2$ . Only $x=4$ survives; $x=-2$ is extraneous.

Practice problems

1.Solve $3^{2x-1} = 27$ .

2.Solve $\log_5(2x+3) = 2$ .

3.Solve $4^x = 2^{x+3}$ .

4.Solve $\log_3(x+2) - \log_3(x-2) = 1$ .

Exponential & Log Equations

Matching bases

Log properties

Practice problems