Applications of Logarithms
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Applications of Logarithms — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionCompound interest applicationsAssessment Practice
8 marks~12 minCriterion A
A savings account pays compound interest at a fixed annual rate. The table below shows the account balance at the end of each of the first four years.

Year1234
Balance (USD)1050.001102.501157.631215.51


The initial deposit made at the start of Year 1 (before any interest) is 1000 USD.
a
Show that the annual interest rate is 5%, and write down the balance at the end of Year 5. [2]
b
Find a formula for the balance BnB_n (in USD) at the end of year nn, where nn is a positive integer. Hence determine the first year in which the balance exceeds 1500 USD. Give your answer to the nearest dollar. [4]
c
A second account offers a one-time bonus of 150 USD added to the initial deposit of 1000 USD, but pays only 3% annual compound interest. Advise a customer, with supporting calculations, which account they should choose if their goal is to maximise their balance at the end of Year 10. [2]
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2QuestionCompound interest applicationsAssessment Practice
8 marks~12 minCriterion D
A technology company tracks the value of a server purchased for 8000 dollars. The value VV (in dollars) after tt years is modelled by
V(t)=8000×(0.78)t.V(t) = 8000 \times (0.78)^{t}.

Year (tt)024
Predicted VV (USD)800048672961
Actual VV (USD)800052003400
a
Calculate the percentage error between the predicted and actual values at t=2t = 2, giving your answer correct to 1 decimal place.
percentage error=∣predicted−actual∣actual×100%\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\% [2]
b
Determine whether the model is acceptable at t=4t = 4, given that the company requires the percentage error to be less than 15%. Show all working. [2]
c
A financial analyst suggests replacing the model with V(t)=8000×rtV(t) = 8000 \times r^{t}, where rr is chosen so that the predicted value exactly matches the actual value of 5200 dollars at t=2t = 2. Find the value of rr, correct to 3 significant figures. Justify whether this revised model is appropriate for predicting the server's long-term value, referring to the behaviour of the model for large values of tt. [4]
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3QuestionpH scale (log scale)Assessment Practice
10 marks~15 minCriterion C
A chemist models the acidity of water samples using the formula
pH=−log⁡10[H+]\text{pH} = -\log_{10}[\text{H}^+]
where [H+][\text{H}^+] is the hydrogen ion concentration in mol/L.

Three water samples have the following measured [H+][\text{H}^+] values and recorded actual pH values from a calibrated probe.

Sample A: [H+]=4.2×10−5[\text{H}^+] = 4.2 \times 10^{-5} mol/L, predicted pH =p= p, actual pH =4.21= 4.21

Sample B: [H+]=3.0×10−7[\text{H}^+] = 3.0 \times 10^{-7} mol/L, predicted pH =q= q, actual pH =6.84= 6.84

Sample C: [H+]=8.5×10−10[\text{H}^+] = 8.5 \times 10^{-10} mol/L, predicted pH =r= r, actual pH =9.04= 9.04
a
Calculate the predicted pH values pp, qq, and rr for samples A, B, and C respectively. Give each answer to 2 decimal places. [3]
b
The percentage error between the predicted and actual pH is defined as
percentage error=∣ predicted−actual ∣actual×100%.\text{percentage error} = \frac{|\,\text{predicted} - \text{actual}\,|}{\text{actual}} \times 100\%.
The chemist claims the percentage error for sample C is less than 0.5%0.5\%. Show that this claim is correct. [3]
c
A fourth sample D has an actual pH of 5.605.60. The chemist uses the model to find [H+][\text{H}^+] for sample D, obtaining [H+]≈2.51×10−6[\text{H}^+] \approx 2.51 \times 10^{-6} mol/L, and concludes that sample D is acidic. Justify whether the chemist's conclusion is correct. [4]
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4QuestionpH scale (log scale)Assessment Practice
14 marks~21 minCriterion B

A river is contaminated by an industrial acid spill. Its pH is measured daily at noon, starting at pH 6.8 immediately after the spill. The pH drops by exactly 0.3 each day for the first 5 days. The relationship between pH and hydrogen ion concentration [H+] (in mol/L) is pH = -log10[H+].

a
[3 marks] Calculate the hydrogen ion concentration [H+] for each of the 5 days (days 0 to 4). Present your results in a table with columns for Day, pH, and [H+] (in scientific notation to 3 significant figures).
b
[2 marks] Plot the data from part (a) on a graph of [H+] (y-axis) against Day (x-axis). Draw a smooth curve through the points.
c
[3 marks] The company responsible claims that the pH will continue to drop by 0.3 each day indefinitely. Using your graph or calculations, evaluate the validity of this claim. In your answer, comment on one limitation of modelling the hydrogen ion concentration with a constant daily pH drop.
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