IBDP Maths analysis and approaches Topic: SL 2.9 : Logarithmic functions and their graphs HL Paper 2


All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death of a plant can be determined by measuring the amount of carbon-14 still present in the remains.

The amount, A , of carbon-14 present in a plant t years after its death can be modelled by A = A0e-kt where t ≥ 0 and A0 , k are positive constants.

At the time of death, a plant is defined to have 100 units of carbon-14.

(a) Show that A0 = 100 . [1]

The time taken for half the original amount of carbon-14 to decay is known to be 5730 years.

(b) Show that \(k=\frac{ln 2}{5730}\) [3]

(c) Find, correct to the nearest 10 years, the time taken after the plant’s death for 25 % of the carbon-14 to decay. [3]



(a) When $t=0$, $A=100$, i.e., we have $$\begin{eqnarray} 100 = A_0\text{e}^{-k\left(0\right)} \nonumber \\ A_0 = 100. \end{eqnarray}$$ (b) Let half-life be $t=5730$, then $$\begin{eqnarray} \frac{1}{2} = \text{e}^{-k\left(5730\right)} \nonumber \\ -\ln 2 = -k\left(5730\right) \nonumber \\ k = \frac{\ln 2}{5730}. \end{eqnarray}$$ (c) When $25%$ of carbon-14 has decayed, we are left with $A=75$. Thus, from the graphing calculator, solving $75=100\text{e}^{-\frac{\ln 2}{5730}t}$, we have $t\approx 2380\text{ years}$.


(a)     Find the solution of the equation

\[\ln {2^{4x – 1}} = \ln {8^{x + 5}} + {\log _2}{16^{1 – 2x}},\]

expressing your answer in terms of \(\ln 2\).

(b)     Using this value of x, find the value of a for which \({\log _a}x = 2\), giving your answer to three decimal places.



(a)     rewrite the equation as \((4x – 1)\ln 2 = (x + 5)\ln 8 + (1 – 2x){\log _2}16\)     (M1)

\((4x – 1)\ln 2 = (3x + 15)\ln 2 + 4 – 8x\)     (M1)(A1)

\(x = \frac{{4 + 16\ln 2}}{{8 + \ln 2}}\)     A1

(b)     \(x = {a^2}\)     (M1)

\(a = 1.318\)     A1

Note: Treat 1.32 as an AP.

Award A0 for ±.

[6 marks]


(a) Let a = log2x, b = log2y, c = log2z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of a, b and c.
(b) Let A = log2x, B = log4y, C = log8z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of A, B and C.


(a) \(log_2(\frac{x^3\sqrt{y}}{z^4})=3a+\frac{1}{2}b-4c\).
(b) \(log_2(\frac{x^3\sqrt{y}}{z^4})\)=3A+B-12C\)


Solve the simultaneous equations


x = 1, y = 3 or \(x=\frac{1}{4}, y=\frac{3}{2}\)

Scroll to Top