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TMUA exponentials and logarithms questions
All 23 exponentials and logarithms questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P1 Q7 · P(x) and Q(x) are defined as follows: P(x) = 2^x + 4 Q(x) = 2^(2x-2) - 2^(x+2) + 16 Find the…
- P1 Q15 · The difference between the maximum and minimum values of the function f(x) = a^cos x, where a > 0… hard
- P2 Q6 · Consider the following equation where a is a real number and a > 1: (*) a^x = x Which of the…
2022
2021
- P1 Q4 · Find the minimum value of the function 2^2x - 2^x+3 + 4
- P2 Q14 · Consider the following simultaneous equations, where p is a real number: p,2^x + log₂ y = 2 2^x +…
- P2 Q17 · Consider the following functions defined for x > 1: f(x) = log₂(log₂ √(x)) g(x) = log₂(√(log₂ x))… hard
2020
- P1 Q6 · Find the maximum value of the function f(x) = 15^2x - 4(5^x) + 7
- P1 Q7 · Given that 2^3x = 8^(y+3) and 4^(x+1) = 16^(y+1)8^(y+3) what is the value of x + y?
- P1 Q15 · Find the positive difference between the two real values of x for which (log₂ x)⁴ +…
2019
- P1 Q11 · Find the sum of the real values of x that satisfy the simultaneous equations log₃(xy²) = 1 (log₃…
- P1 Q13 · Find the maximum value of 4^sin x - 4 × 2^sin x + (17)/(4) for real x.
- P1 Q15 · Find the real non-zero solution to the equation 2^(9^x)8^(3^x) = 14
- P2 Q15 · The numbers a, b and c are each greater than 1. The following logarithms are all to the same base:…
2018
- P1 Q14 · The line y = mx + 4 passes through the points (3, log₂ p) and (log₂ p, 4). What are the possible…
- P1 Q15 · Find the sum of the real solutions of the equation: 3^x - (√(3))^x+4 + 20 = 0
2017
- P1 Q14 · The solution of the simultaneous equations 2^x + 3 × 2^y = 3 2^2x - 9 × 2^2y = 6 is x = p, y = q.… hard
- P1 Q18 · The graph of y = log₁₀ x is translated in the positive y-direction by 2 units. This translation is…
- P2 Q7 · The graphs of two functions are shown here: y = a^x is shown with a solid line, where a is a…
- P2 Q8 · Which one of the following numbers is smallest in value?
2016
- P1 Q11 · The real roots of the equation 4^2x + 12 = 2^2x+3 are p and q, where p > q. The value of p - q can…
- P1 Q16 · Given the simultaneous equations log₁₀ 2 + log₁₀(y-1) = 2log₁₀ x log₁₀(y + 3 - 3x) = 0 the values…
- P2 Q7 · The four real numbers a, b, c, and d are all greater than 1. Suppose that they satisfy the…