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VICMath MethodsQuick questions
Unit 3
Quick questions on Solving polynomial, exponential, logarithmic and circular equations: VCE Math Methods Unit 3
14short 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?Show answer
If both sides can be written with the same base, equate exponents.
What is using the change of base?Show answer
If different bases appear, convert using $a^x = e^{x \ln a}$ (or take $\ln$ of both sides).
What is substitution trick for quadratic-in-exponential?Show answer
Equations like $e^{2x} - 5 e^x + 6 = 0$ become quadratics under $u = e^x$.
What is combine using log laws?Show answer
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?Show answer
Once you have $\ln(\text{expression}) = c$, exponentiate both sides: expression $= e^c$. Then solve the resulting algebraic equation.
What is check for spurious solutions?Show answer
Log arguments must be positive. After solving, substitute back into the original equation and reject any solutions that make a log argument zero or negative.
What is one basic period?Show answer
To solve $\sin(\theta) = k$ for $\theta$ in a given interval:
What is compound argument?Show answer
For $\sin(b \theta - c) = k$ on $[0, 2\pi]$, let $u = b\theta - c$. The new variable $u$ ranges over $[-c, 2\pi b - c]$. Find all solutions in this extended range, then back-substitute and solve for $\theta$.
What is trig identities for solving?Show answer
The Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ lets you convert between $\sin$ and $\cos$. The substitution $u = \sin(x)$ (or $\cos(x)$) reduces some trig equations to quadratics.
What is dropping solutions when squaring?Show answer
Squaring both sides of an equation can introduce extraneous roots; always substitute back to check.
What is wrong interval for a substituted variable?Show answer
When $u = 2x$ and $x \in [0, 2\pi]$, $u$ ranges over $[0, 4\pi]$, not $[0, 2\pi]$. Doubling the argument doubles the number of solutions.
What is ignoring the domain on log equations?Show answer
Spurious solutions from log equations always come back to argument-positivity. Always check.
What is treating $\log $ as $\log + \log $?Show answer
This is false. Only $\log(ab) = \log(a) + \log(b)$.
What is computing decimals on Paper 1?Show answer
Exact answers like $\frac{\ln 7}{\ln 3}$, $\ln 2$, or $\pi/6$ are required.