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VICMath MethodsQuick questions

Unit 3

Quick questions on Solving polynomial, exponential, logarithmic and circular equations: VCE Math Methods Unit 3

10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is standard form, single base?
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If both sides can be written with the same base, equate exponents.
What is using the change of base?
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If different bases appear, convert using ax=exln⁡aa^x = e^{x \ln a} (or take ln⁡\ln of both sides).
What is substitution trick for quadratic-in-exponential?
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Equations like e2x−5ex+6=0e^{2x} - 5 e^x + 6 = 0 become quadratics under u=exu = e^x.
What are combine using log laws?
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If the equation contains multiple log terms, combine using the product, quotient and power laws into a single log.
What is exponentiate to remove the log?
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Once you have ln⁡(expression)=c\ln(\text{expression}) = c, exponentiate both sides: expression =ec= e^c. Then solve the resulting algebraic equation.
What is trig identities for solving?
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The Pythagorean identity sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1 lets you convert between sin⁡\sin and cos⁡\cos. The substitution u=sin⁡(x)u = \sin(x) (or cos⁡(x)\cos(x)) reduces some trig equations to quadratics.
What is wrong interval for a substituted variable?
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When u=2xu = 2x and x∈[0,2π]x \in [0, 2\pi], uu ranges over [0,4π][0, 4\pi], not [0,2π][0, 2\pi]. Doubling the argument doubles the number of solutions.
What is q1?
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Solve 52x−1=255^{2x - 1} = 25 for xx. [2 marks]
What is q2?
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Solve log⁡3(x)+log⁡3(x−2)=1\log_3(x) + \log_3(x - 2) = 1 for xx. [3 marks]
What is q3?
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Solve cos⁡(2x)=12\cos(2x) = \frac{1}{2} for x∈[0,2π]x \in [0, 2\pi]. [3 marks]

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