Pure: Exponentials and Logarithms
A-level Pure Mathematics' exponentials and logarithms cover the functions a^x and e^x, their inverse logarithms (including the natural logarithm ln), and how to solve equations and model real-world growth or decay using them. Students must know the laws of logarithms, how to solve equations by taking logs of both sides, and how these functions link to sequences, differentiation and integration.
Method
- Learn the laws of logarithms: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), and log(a^n) = n*log(a), and apply them to combine or split logarithmic expressions.
- To solve an equation like a^x = b, take logs (or ln) of both sides, use log(a^x) = x*log(a) to bring the power down, and rearrange to make x the subject.
- To solve an equation containing logs, combine every log term into a single log first, then remove the log by rewriting the equation in index form.
- Always state any restrictions on the domain (for example x > 0 inside a log) and reject any solution that falls outside them.
- For exponential growth or decay models such as N = N0*e^(kt), substitute given information to find the unknown constant, then use the model to answer questions about specific times or values.
- Remember that y = ln(x) and y = e^x are inverse functions and reflections of each other in the line y = x, which helps with sketching graphs and finding asymptotes.
Worked example
Solve the equation ln(2x - 1) = 3, giving your answer to 3 significant figures.
- Rewrite the equation in index (exponential) form: 2x - 1 = e^3.
- Calculate e^3 = 20.0855 (4dp).
- Add 1 to both sides: 2x = 21.0855.
- Divide by 2: x = 10.5427.
- Final answer: x = 10.5 (3 significant figures).
Practice questions
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Q1Write log_2(32) = 5 in index form.Show answer
Answer: 2^5 = 32.
Q2Solve 3^x = 81 without a calculator.Show answer
Answer: x = 4 (since 81 = 3^4).
Q3Simplify log_4(20) - log_4(5), giving your answer as an integer.Show answer
Answer: 1 (log_4(20/5) = log_4(4)).
Q4Solve log_3(2x + 1) = 4.Show answer
Answer: x = 40 (2x + 1 = 3^4 = 81).
Q5Solve 5^x = 40, giving your answer to 3 significant figures.Show answer
Answer: x = 2.29 (x = ln(40)/ln(5)).
Q6Solve ln(x) + ln(x - 3) = ln(4), stating why one solution must be rejected.Show answer
Answer: x = 4 (reject x = -1 since ln(x) needs x > 0).
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
Solve 2^x = 50, giving your answer to 3 significant figures.
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A population of bacteria is modelled by P = 500*e^(0.08t), where P is the number of bacteria and t is the time in hours. Find the time taken for the population to reach 2000, giving your answer to 3 significant figures.
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The value, V pounds, of a delivery van t years after purchase is modelled by V = 18000*e^(-kt), where k is a positive constant. The van's value has fallen to 11000 pounds after 4 years. Show that k = 0.25*ln(18/11), giving the value of k to 3 significant figures, and use this model to find the value of the van after 10 years, giving your answer to the nearest pound.
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Free printable worksheet
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