Questions / ESAT / Maths 2: Exponentials & logs Engineering and Science Admissions Test
ESAT Maths 2 Exponentials & logs questions
3 original ESAT Maths 2 exponentials & logs questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.
Methods on this page: Solve an exponential equation as a quadratic Combine logarithms, then check the domain
Question 1 ESAT Maths 2 Warm-up
What is the sum of the real solutions of
9 x − 12 × 3 x + 27 = 0 9^x - 12 \times 3^x + 27 = 0 9 x − 12 × 3 x + 27 = 0 ?
A 1 1 1 B 2 2 2 C 3 3 3 D 4 4 4 E 5 5 5 F 6 6 6 G 12 12 12 Show the answer and worked solution
Answer: C · Method: Solve an exponential equation as a quadratic
Let u = 3 x u = 3^x u = 3 x 9 x = u 2 9^x = u^2 9 x = u 2 u 2 − 12 u + 27 = 0 u^2 - 12u + 27 = 0 u 2 − 12 u + 27 = 0 u = 9 u = 9 u = 9 u = 3 u = 3 u = 3 3 x = 9 3^x = 9 3 x = 9 x = 2 x = 2 x = 2 3 x = 3 3^x = 3 3 x = 3 x = 1 x = 1 x = 1 3 3 3 Why the tempting wrong answers are wrong
G: That's the sum of the values of 3 x 3^x 3 x x x x Try it interactively →
Question 2 ESAT Maths 2 Standard
What is the sum of all the real solutions of
log 3 ( x + 21 ) + log 3 ( x − 5 ) = 3 \log_{3}(x + 21) + \log_{3}(x - 5) = 3 log 3 ( x + 21 ) + log 3 ( x − 5 ) = 3 ?
A − 22 -22 − 22 B − 16 -16 − 16 C 6 6 6 D 7 7 7 E 27 27 27 Show the answer and worked solution
Answer: C · Method: Combine logarithms, then check the domain
Combine the logs: log 3 ( ( x + 21 ) ( x − 5 ) ) = 3 \log_{3}\big((x + 21)(x - 5)\big) = 3 log 3 ( ( x + 21 ) ( x − 5 ) ) = 3 ( x + 21 ) ( x − 5 ) = 27 (x + 21)(x - 5) = 27 ( x + 21 ) ( x − 5 ) = 27 Expanding: x 2 + 16 x − 132 = 0 x^2 + 16x - 132 = 0 x 2 + 16 x − 132 = 0 x = 6 x = 6 x = 6 x = − 22 x = -22 x = − 22 Both brackets must be positive. At x = − 22 x = -22 x = − 22 x − 5 = − 27 < 0 x - 5 = -27 < 0 x − 5 = − 27 < 0 x = 6 x = 6 x = 6 Why the tempting wrong answers are wrong
A: At x = − 22 x = -22 x = − 22 x − 5 x - 5 x − 5 B: One of the roots makes a logarithm of a negative number. Always check the domain.Try it interactively →
Question 3 ESAT Maths 2 Standard
What is the sum of the real solutions of
4 x − 4 × 2 x − 32 = 0 4^x - 4 \times 2^x - 32 = 0 4 x − 4 × 2 x − 32 = 0 ?
A 1 1 1 B 2 2 2 C 3 3 3 D 4 4 4 E 5 5 5 F 6 6 6 G 7 7 7 Show the answer and worked solution
Answer: C · Method: Solve an exponential equation as a quadratic
Let u = 2 x u = 2^x u = 2 x 4 x = u 2 4^x = u^2 4 x = u 2 u 2 − 4 u − 32 = 0 u^2 - 4u - 32 = 0 u 2 − 4 u − 32 = 0 u = 8 u = 8 u = 8 u = − 4 u = -4 u = − 4 2 x = 8 2^x = 8 2 x = 8 x = 3 x = 3 x = 3 2 x = − 4 2^x = -4 2 x = − 4 2 x > 0 2^x > 0 2 x > 0 3 3 3 Why the tempting wrong answers are wrong
D: That's the sum of the values of 2 x 2^x 2 x x x x E: 2 x 2^x 2 x 2 x = − 4 2^x = -4 2 x = − 4 Try it interactively →
There are 23 exponentials & logs questions on phimaths. Free, no sign-up. Get one wrong and you get a new question that needs the same method.
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