Solving exponential equations
If both sides can be written with the same base, equate the powers. For example 2ˣ = 32 = 2⁵, so x = 5. When the bases differ, take logs of both sides and use the power law.
Key idea
aˣ = b ⇒ x = (lg b) / (lg a). The natural log ln is log to base e; ln x and eˣ are inverse operations, so ln(eˣ) = x and e^(ln x) = x.
Worked example
Solve 5ˣ = 20 to 2 decimal places. Take logs: x lg 5 = lg 20, so x = lg 20 ÷ lg 5 = 1.30103 ÷ 0.69897 = 1.86 (to 2 d.p.).
Graphs
y = aˣ (a > 1) passes through (0, 1), increases and has the x-axis (y = 0) as a horizontal asymptote. Its inverse y = loga x passes through (1, 0) and has the y-axis (x = 0) as a vertical asymptote.
Remember
- You can only take log of a positive number.
- aˣ > 0 for all x, so it never touches the x-axis.
- To solve loga(f(x)) = k, rewrite as f(x) = aᵏ.