Symbolic manipulations
A base-5 log, a triple angle and a quadratic in 3ˣ
- Units 2 and 3
- 6 points
- About 18 minutes
You solve exponential, logarithmic and trigonometric equations by hand and rewrite log, exponential and trig expressions in equivalent forms, giving exact answers and showing your work. No calculator. On the exam: Question 4 of 4. Part B of the free-response section: 2 questions in 35 minutes, no calculator allowed.
The question
No calculator is allowed for this question. Angle measures are in radians. Unless otherwise specified, the domain of a function f is the set of all real numbers x for which f(x) is a real number, and solutions to equations must be real numbers. Give the exact value of any expression that can be evaluated without a calculator (for example, log₂ 8, cos(π/2) and arcsin(1)). Unless otherwise specified, combine terms using algebra and the rules for exponents and logarithms (for example, write x²·x³ as x⁵, and ln 3 + ln 5 as ln 15). Show the work that leads to each answer.
Suggested time: 18 minutes
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Part (a)
2 pointsThe functions g and h are given by g(x) = log₅(2x + 1) + 3 and h(x) = 2 cos(3x). (i) Solve g(x) = 1 for values of x in the domain of g. (ii) Solve h(x) = −√2 for values of x in the interval [0, π/2].
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Part (b)
2 pointsThe functions j and k are given by j(x) = ln(x² − 9) − ln(x − 3) − ln 2 and k(x) = csc x − cot x · cos x. (i) Rewrite j(x) as a single natural logarithm without negative exponents in any part of the expression. Your result should be of the form ln(expression). (ii) Rewrite k(x) so that the only trigonometric function in it is sin x, and sin x is used just once.
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Part (c)
2 pointsThe function m is given by m(x) = 3^(2x) − 3^(x + 1) − 10. Find all input values in the domain of m that yield an output value of 0.
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