The big idea: If both sides can be written with the same base, then the exponents must be equal.
Worked example
Solve 2x+1 = 16.
Step by step
- Rewrite 16 as a power of 2.
- Rewrite the equation with the common base.
- Since the bases are equal, the exponents must be equal too — drop the common base.
- Solve.
Final answer
x = 3
Use common bases when you can: This is often the fastest method and avoids unnecessary calculator work.
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When bases do not match: If you cannot rewrite both sides with a common base, use logarithms to bring the exponent down.
Worked example
Solve 3x = 20.
Step by step
- Take logs of both sides.
- Use the power law.
- Solve for x.
Final answer
x ≈ 2.727
Any consistent log base works: You can use log or ln, as long as you use the same one on both sides.
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The big idea: A logarithmic equation can often be rewritten as an exponential equation.
Worked example
Solve log2 x = 5.
Step by step
- Rewrite in exponential form.
- Calculate.
Final answer
x = 32
Second example
Solve log10 x = 2.3.
Step by step
- Rewrite in exponential form.
- Use a calculator.
Final answer
x ≈ 199.53
Logs need positive inputs: log x only makes sense when x > 0 — you cannot take the logarithm of 0 or a negative number in the real numbers.
So if you solve a log equation and get something like x = −100, reject it: log(−100) is undefined.
👉 Always check your answer back in the original equation — negative or zero answers should be rejected when the unknown sits inside a log.
🎯 IB-style worked example
Two GDC keys you'll use here:
- LOG → use for part (a) when concentration is given and you want pH.
- 2nd + LOG → 10ˣ; use for parts (b) and (c) to undo the log and find [H⁺].
Scenario: The pH of a solution tells you how acidic it is. The smaller the pH, the stronger the acid.
pH is linked to the hydrogen ion concentration [H⁺] (measured in moles per litre) by the formula:
pH = −log₁₀[H⁺]
Examples: pH 2 (lemon juice) is much more acidic than pH 5 (black coffee), which is more acidic than pH 7 (water).
Part (a) — find pH from concentration
Using pH = −log₁₀[H⁺]:
(a) Find the pH when [H⁺] = 0.00042 mol L⁻¹. [2 marks]
Step by step
- Substitute [H⁺] = 0.00042 into the formula:
- Evaluate log(0.00042) on the GDC. The minus sign in the formula flips the negative log into a positive pH:
- Round to 3 s.f.:
Final answer
pH ≈ 3.38
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (b) — find concentration from pH
Using the same formula:
(b) Find [H⁺] when pH = 6.8. Give your answer in the form a × 10ᵏ. [2 marks]
Step by step
- Substitute pH = 6.8 into the formula:
- Multiply both sides by −1 to isolate the log:
- Undo the log with 10ˣ:
Final answer
[H⁺] ≈ 1.58 × 10⁻⁷ mol L⁻¹
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (c) — compare two acids
A juice has pH 2.3 and tea has pH 4.9.
(c) How many times more acidic is the juice than the tea? [3 marks]
Step by step
- Convert both pH values to [H⁺] using 10ˣ:
- Divide stronger by weaker to get the ratio. Use the index law :
- Evaluate on the GDC:
- Write the answer in context:
Final answer
Juice is ≈ 398 times more acidic than tea.
Exam tips for pH questions:
- The minus sign matters. pH = −log[H⁺] gives a positive pH because [H⁺] is small (less than 1), so log[H⁺] is negative.
- Use the (−) key, not the subtract key, for negative exponents like 10−6.8.
- For 'how many times stronger' questions, divide the larger concentration by the smaller — the ratio comes out clean using the index law 10a ÷ 10b = 10a−b.
- Always keep units — concentration is in mol L⁻¹ (moles per litre).
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
What you'll practise:
- Plug numbers into a log model to find a missing constant.
- Substitute a data pair to find another constant.
- Find the smallest and biggest possible value of N.
- Turn a rate into a waiting time using 1/N.
Two GDC keys you'll use:
- LOG → evaluates log₁₀ (use for part (a) to find a).
- 2nd + LOG → 10ˣ (use for part (b) to convert a into b).
✏️ IB-style worked example
Scenario: Background: Earthquake size is described by a magnitude M on the Richter scale, which goes from about 0 (barely detectable) up to roughly 8 (catastrophic). Stronger quakes are much rarer than weaker ones.
For a given region, the average number of earthquakes per year, N, with magnitude at least M is modelled by log₁₀ N = a − M for some constant a.
Suppose that in one region there are, on average, 50 earthquakes per year with magnitude at least M = 3.
Part (a) — find a
Using log₁₀ N = a − M with N = 50 and M = 3:
(a) Find a. [2 marks]
Step by step
- Substitute N = 50 and M = 3 into the model:
- Solve for a by adding 3 to both sides:
- Type into the GDC with LOG, then round to 3 s.f.:
Final answer
a ≈ 4.70
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (b) — rewrite as N = b / 10M and find b
Using the same model:
(b) Show that the model can be written as N = b / 10M and find b. [2 marks]
Step by step
- The question gives you the target form N = b / 10M and the data pair N = 50 when M = 3. So just substitute straight in:
- Multiply both sides by 10³ to isolate b:
- Write in standard form (the answer to 3 s.f.):
Final answer
b = 5.0 × 10⁴
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (c) — range of N when 0 < M < 8
(c) For 0 < M < 8 (covering the full Richter range from barely felt to catastrophic), find the range of N. [2 marks]
Use N = b / 10M with b = 5.0 × 10⁴ (= 50000).
Step by step
- Pick the two endpoints of M. The Richter scale runs from 0 to 8, so try M just above 0 and M just below 8. Use the model from part (b):
- Endpoint M = 0. When M is tiny, 10M is close to 1, so N is largest:
- Endpoint M = 8. When M is big, 10M is huge, so N is smallest:
- Write the range. N can be anywhere between these two values (use strict < because M is strictly between 0 and 8):
Final answer
0.0005 < N < 50000
Part (d) — expected waiting time for M ≥ 5.8
Using the same model:
(d) The largest recent earthquake had M = 5.8. The expected waiting time (in years) for the next earthquake of at least this size is 1/N. Find this time to the nearest year. [2 marks]
Step by step
- Recall the model from part (b) with b = 50000:
- Substitute M = 5.8 into the model:
- Take the reciprocal to get the waiting time in years:
- Round to the nearest year:
Final answer
≈ 13 years
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Exam tips for log-model questions:
- Find 'a' first — every other part needs it.
- Keep a unrounded. Use Ans on the GDC to avoid rounding errors between parts.
- N decreases as M increases. Plug in the endpoints to get the range.
- Waiting time = 1/N. Always take the reciprocal after you find N — never invert M.