Algebraic manipulation of polynomial, exponential and logarithmic expressions, including index laws, logarithm laws, factorisation, and the solution of linear, quadratic, polynomial, exponential and logarithmic equations

What this dot point is asking

VCAA wants you to manipulate algebraic expressions involving indices and logarithms, factorise polynomial expressions, and solve linear, quadratic, polynomial, exponential and logarithmic equations. The dot point builds the algebraic fluency Unit 3 / 4 will require.

Index laws

For real $a, b$ and integer or rational $m, n$:

Logarithm laws

For $a > 0, a \neq 1$ and positive $m, n$:

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Change of base.

So $\log_2 5 = \log_{10} 5 / \log_{10} 2 \approx 2.32$.

Inverse relationship. $\log_a(a^x) = x$ and $a^{\log_a x} = x$.

Factorisation techniques

Common factor. $6 x^3 - 9 x^2 = 3 x^2 (2 x - 3)$.

Grouping. $x^3 + 2 x^2 - x - 2 = x^2(x + 2) - (x + 2) = (x + 2)(x^2 - 1) = (x + 2)(x - 1)(x + 1)$.

Quadratic factorisation. $x^2 + 5x + 6 = (x + 2)(x + 3)$. Use sum-and-product (looking for two numbers that sum to the middle coefficient and multiply to the constant).

Quadratic formula. $a x^2 + b x + c = 0$ has solutions $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$.

Difference of squares. $a^2 - b^2 = (a - b)(a + b)$.

Sum and difference of cubes. $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$; $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Solving equations

Linear. Single step or simple multi-step manipulation.

Quadratic. Factor first, then use the null factor law ($AB = 0 \implies A = 0$ or $B = 0$). Or use the quadratic formula.

Polynomial. Factor where possible. Look for rational roots first; then use polynomial division or grouping.

Exponential. Bring to common base if possible, then equate exponents. Otherwise take logarithms.

Example: $5^x = 17$. Take $\log_{10}$: $x \log_{10} 5 = \log_{10} 17$, so $x = \log_{10} 17 / \log_{10} 5 \approx 1.76$.

Logarithmic. Combine logs using laws; convert to exponential form. Always check domain (logs require positive arguments).

Example: $\log_3(x + 5) = 2$. Convert: $x + 5 = 3^2 = 9$, so $x = 4$. Check: $\log_3(9) = 2$. Confirmed.

Worked example: solving an exponential equation

Solve $4^x - 3 \cdot 2^x + 2 = 0$.

Substitute $u = 2^x$. Then $4^x = (2^x)^2 = u^2$.

Equation becomes $u^2 - 3u + 2 = 0$, factoring as $(u - 1)(u - 2) = 0$.

So $u = 1$ or $u = 2$, i.e. $2^x = 1$ or $2^x = 2$, giving $x = 0$ or $x = 1$.

Worked example: simultaneous equations

$3 x + 2 y = 14$ and $x - y = 3$.

From the second: $x = y + 3$. Substitute: $3(y + 3) + 2 y = 14$, so $5 y + 9 = 14$, $y = 1$, then $x = 4$.

Common errors

Index law inversion. $a^{-n} = 1/a^n$, not $-a^n$.

Log of a negative. Logs of negative numbers and zero are undefined in the real numbers. Always check.

Wrong factorisation. $a^2 - b^2 = (a - b)(a + b)$ is the difference of squares. $a^2 + b^2$ does not factor over the reals.

Forgetting both roots. Quadratic equations have (in general) two solutions; report both.

Missing the domain check on log equations. $x = -4.46$ in the worked example would be rejected because it violates the log argument constraint.

In one sentence

Unit 1 algebra establishes fluency with index laws, logarithm laws (including change of base), polynomial factorisation (common factor, grouping, quadratic, difference of squares, sum/difference of cubes) and the solution of linear, quadratic, polynomial, exponential and logarithmic equations; the most-tested moves are bringing exponentials to common base, combining logs before solving, and the mandatory domain check on logarithmic solutions.