IB Maths AA SL · Unit 1: Number and Algebra
IB Maths AA SL Exponentials and Logarithms Questions
Exam-style IB Maths AA SL exponentials and logarithms questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 44 questions
- Paper 1: 36
- Paper 2: 8
- 8 easy
- 19 medium
- 10 hard
- 2 very hard
- 5 starter
- 3 worked examples
Practise Exponentials and Logarithms questions →
AA SL formula booklet
What you need to know
SL AA tests logs abstractly: solve log(x+1) + log(x-1) = 1. Master the three log laws (product, quotient, power) and change of base. Exponent and logarithm laws overview →
What's examined in AA SL exponentials and logarithms
The question bank covers these exponentials and logarithms question types (number of questions in brackets):
- Solving Exponential and Logarithmic Equations (27)
- Exponential and Logarithm Properties (10)
- Exponential and Logarithm Functions and Models (7)
Key formulas
- Exponents & logs (change of base)
- \(\log_a x = \dfrac{\log_b x}{\log_b a}\)
In the same notation as the IB formula booklet. All AA SL formulas →
Exponentials and Logarithms worked examples
Worked example 1: Applying logarithmic laws · easy
Solve the logarithmic equation $\log_6 3 + \log_6(2x) = 2 - \log_6 12$.
1. Group all logarithmic terms on the left side of the equation: $\log_6 3 + \log_6(2x) + \log_6 12 = 2$.
2. Apply the product law of logarithms ($\log a + \log b = \log(ab)$) to combine the terms: $\log_6(3 \times 2x \times 12) = 2$.
3. Simplify the argument of the logarithm: $\log_6(72x) = 2$.
4. Convert the logarithmic equation into its exponential form: $6^2 = 72x$.
5. Simplify the constant: $36 = 72x$.
6. Solve for $x$: $x = \frac{36}{72} = \mathbf{0.5}$.
Examiner tip: A very common trap is incorrectly "adding" the arguments instead of multiplying them when applying the log addition rule. Remember, logs are exponents, so adding them corresponds to multiplying the bases!
Worked example 2: Change of base formula · medium
Use the change of base formula to solve the equation $\log_4 x + \log_{16} x = 3$.
1. Apply the change of base formula to express $\log_{16} x$ in base 4: $\log_{16} x = \frac{\log_4 x}{\log_4 16}$.
2. Evaluate the denominator since $4^2 = 16$: $\log_4 16 = 2$.
3. Substitute this back into the original equation: $\log_4 x + \frac{\log_4 x}{2} = 3$.
4. Factor out $\log_4 x$: $1.5 \log_4 x = 3 \implies \frac{3}{2} \log_4 x = 3$.
5. Isolate the logarithm: $\log_4 x = 3 \times \frac{2}{3} = 2$.
6. Convert to exponential form to solve: $x = 4^2 = \mathbf{16}$.
Examiner tip: When changing bases, students often flip the fraction upside down. Always remember: the original base goes to the bottom ($\log_b a = \frac{\log_c a}{\log_c b}$).
Worked example 3: Hidden quadratics in exponentials · hard
Solve the exponential equation $4^x - 3 \times 2^{x+2} = 64$.
1. Rewrite all exponential terms to have a common base of 2: $(2^2)^x - 3(2^x \times 2^2) = 64$.
2. Apply exponent laws to simplify: $(2^x)^2 - 3(4 \times 2^x) = 64 \implies (2^x)^2 - 12(2^x) - 64 = 0$.
3. Substitute a dummy variable $u = 2^x$ to reveal the hidden quadratic: $u^2 - 12u - 64 = 0$.
4. Factorise the quadratic equation: $(u - 16)(u + 4) = 0$, giving solutions $u = 16$ and $u = -4$.
5. Substitute back $u = 2^x$: $2^x = 16$ and $2^x = -4$.
6. Solve for $x$. Since $2^x = -4$ has no real solutions (exponential functions are strictly positive), we only solve $2^x = 16 \implies \mathbf{x = 4}$.
Examiner tip: A frequent mistake is rewriting $2^{x+2}$ as $2 \times 2^x$. The correct expansion using index laws is $2^x \times 2^2 = 4 \times 2^x$. Also, never forget to explicitly reject the negative root!
Try these IB Maths AA SL exponentials and logarithms questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 5 marks · Paper 1
Find the exact value of each of the following:
\(\log_2 16\)
\(\log 25 + \log 4\)
\(\log_5 500 - \log_5 4\)
Question 2 · medium · 6 marks · Paper 2
Let \(f(x) = \ln(x+2)\) for \(x > -2\).
Find the exact coordinates of the \(x\)-intercept and the \(y\)-intercept.
State the equation of the vertical asymptote to the graph of \(f\).
The graph of \(y = f(x)\) intersects with its inverse, \(y = f^{-1}(x)\), twice. Using your GDC, find the coordinates of these two points of intersection.
Question 3 · hard · 8 marks · Paper 2
Let \(f(x) = 0.5e^{2x} + 1\).
Sketch the graph of \(y = f(x)\) for \(-2 \le x \le 1\). Clearly label the \(y\)-intercept and draw the horizontal asymptote.
The inverse of \(f\) can be written in the form \(f^{-1}(x) = A \ln(b(x - c))\). Find the values of \(A\), \(b\), and \(c\).
All 44 exponentials and logarithms questions with mark schemes →
FAQ
How many IB Maths AA SL exponentials and logarithms questions are there?
There are 44 exam-style exponentials and logarithms questions in the AA SL question bank (Paper 1: 36 · Paper 2: 8), graded 8 easy, 19 medium, 10 hard, 2 very hard, 5 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is exponentials and logarithms on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 36 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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