Make the bases the same: The cleanest exponential equations let you write both sides as powers of the same base — then the exponents must be equal.
IB-style question — match the base
Solve 4ˣ = 8.
Step by step
- Write both sides as powers of 2.
- Same base ⇒ equal exponents.
- Solve.
Final answer
x = 3/2.
IB-style question — a shifted power
Solve 9x+1 = 27.
Step by step
- Both are powers of 3.
- Equate exponents and solve.
Final answer
x = 1/2.
If the bases won't match: Not every equation has a common base (e.g. 5ˣ = 20). Then take logs of both sides — the next section.
How do we get x out of an exponent?: If x is in an exponent, use a logarithm. Logs bring exponents down to the front, where you can solve for x.
Why does ln work with e?: ln and e are opposite operations, just like squaring and square roots are opposites.
√(x²) = x
ln(eˣ) = x
So whenever you see e raised to a power containing x, ln is usually the quickest way to get x out of the exponent.
IB-style question — take logs
Solve 5ˣ = 20.
Step by step
- Take logs of both sides.
- The power law brings x to the front.
- Divide by log 5.
Final answer
x = log 20 / log 5 (≈ 1.86).
IB-style question — exact form with e
Solve 90e−0.5x = 2, giving the exact value of x.
Step by step
- Get the e-term on its own.
- Use ln to undo the e.
- ln and e cancel, leaving just the exponent.
- Rewrite 1/45 as 45⁻¹.
- Use the power law.
- Divide by −0.5.
Final answer
x = 2 ln(45).
Exact vs decimal: If the question says "exact", leave your answer as a logarithm (for example 2 ln 45).
If the question does not say exact, give a decimal answer, usually to 3 significant figures.
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Undo the log: A single logarithm equal to a number? Convert to exponential form: loga(expr) = c means expr = aᶜ.
IB-style question — solve a log equation
Solve ln(x² − 16) = 0.
Step by step
- Convert to exponential form (base e).
- Solve for x².
- Both roots keep x² − 16 > 0, so both are valid.
Final answer
x = ±√17.
IB-style question — find an inverse
Find the inverse of f(x) = log₂(8x).
Step by step
- Write y = f(x), then swap x and y.
- Convert to exponential form.
- Solve for y (8 = 2³).
Final answer
f⁻¹(x) = 2x − 3.
Two logs? Make them one: Two logs in one equation? Combine them into a single log first (product or quotient law), then convert and solve — and reject any root that makes a log's argument zero or negative.
IB-style question — combine then solve
Solve log₂ x + log₂(x − 2) = 3.
Step by step
- Product law combines the two logs.
- Convert to exponential form.
- Solve the quadratic.
- Reject x = −2 (need x − 2 > 0).
Final answer
x = 4 (x = −2 rejected).
IB-style question — find the base
Given that logk 81 = 4, find the base k.
Step by step
- Convert to exponential form.
- Take the positive 4th root (a base is positive).
Final answer
k = 3.
Always check the argument: Every logarithm needs a positive argument. After solving, discard any value that makes an argument ≤ 0 — a classic place to lose a mark.
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Graph both sides and find the crossing: On Paper 2, when an equation won't solve neatly by hand, graph each side and read off where they cross.
IB-style question — solve on the GDC
Solve 2ˣ = 30, giving x to 3 significant figures.
Step by step
- By hand this needs logs: x = log₂30 = ln 30 ÷ ln 2.
- On the GDC, graph both sides and intersect (shown below).
Final answer
x ≈ 4.91.
GDC tip (Paper 2): Enter Y₁ = left side and Y₂ = right side, then 2nd → TRACE → 5: intersect.
Pick a window that actually shows the crossing, and watch for more than one intersection.