Flashcards · 13 cards

Indices, logarithms & surds

Say the answer to yourself, then check. Cards you know come back less and less often; cards you don't come back tomorrow.

Know it
If $a^p = a^q$ , what follows?
Answer
$p = q$ (for $a > 0$ , $a \ne 1$ ). So write both sides as powers of the same base, then set the indices equal.
From the lesson: Indices and exponential equations
Rule
$(2^3)^{x - 1} = \,?$
Answer
$2^{3(x - 1)} = 2^{3x - 3}$ : multiply the indices, bracketing the whole index.
A power of a powerMultiply the indices: (2³)ˣ⁻¹ = 2³⁽ˣ⁻¹⁾ = 2³ˣ⁻³
From the lesson: Indices and exponential equations
Rule
How do you simplify $5^{3x - 4} + 5^{3x - 3}$ ?
Answer
Take out the smallest power: $5^{3x - 4}(1 + 5) = 6 \times 5^{3x - 4}$ . Indices don't add across a sum.
Take out the smallest powerEvery other term is it times a number
From the lesson: Indices and exponential equations
Rule
What does $\log_a x = n$ mean?
Answer
$a^n = x$ : the log is the power.
The log is the powerlogₐ x = n means aⁿ = x
From the lesson: Logarithms at Further Maths depth
Know it
The three laws of logarithms?
Answer
$\log_a xy = \log_a x + \log_a y$ , $\log_a \dfrac xy = \log_a x - \log_a y$ and $\log_a x^n = n\log_a x$ .
From the lesson: Logarithms at Further Maths depth
Rule
Change of base, and what happens when you swap the base and the number?
Answer
$\log_a x = \dfrac{\log x}{\log a}$ . Swapping turns the log upside down: $\log_x a = \dfrac{1}{\log_a x}$ .
Change of baseDivide by the log of the base; swap them and the log turns upside down
From the lesson: Logarithms at Further Maths depth
Know it
Is $\dfrac{\log 16}{\log 2}$ equal to $\log 8$ ?
Answer
No. Write the top as a multiple of the bottom: $\log 16 = 4\log 2$ , so the value is 4.
From the lesson: Logarithms at Further Maths depth
Know it
After solving a log equation, what must you check?
Answer
Put each answer back into every log: a log exists only for a positive number (and a base must be positive and not 1).
From the lesson: Logarithms at Further Maths depth
Rule
$(a + \sqrt b)(a - \sqrt b) = \,?$
Answer
$a^2 - b$ : the surds cancel.
The difference of two squares(a + √b)(a − √b) = a² − b: the surds cancel
From the lesson: Surds at Further Maths depth
Rule
How do you solve an equation like $\sqrt{3x + 1} - \sqrt{2x - 1} = 1$ ?
Answer
Get one root on its own and square, keeping the middle term. Get the remaining root on its own and square again. Then test every answer in the original equation.
Square, then checkBoth roots of the quadratic must be tested in the first equation
From the lesson: Surds at Further Maths depth
Know it
$(1 + \sqrt{2x - 1})^2 = \,?$
Answer
$1 + 2\sqrt{2x - 1} + (2x - 1)$ . Don't forget the middle term.
From the lesson: Surds at Further Maths depth
Which method?
WAEC 2018 · Paper 2 · Q2
Given that $\log_3 x - 3\log_x 3 + 2 = 0$ , find the values of $x$ .
How do you deal with $\log_x 3$ ?
Answer
Swap base and number: $\log_x 3 = \dfrac{1}{\log_3 x}$ . Then let $y = \log_3 x$ to get an equation in $y$ .
From the lesson: Logarithms at Further Maths depth Try the question
Which method?
WAEC 2019 · Paper 2 · Q1
Simplify: $\dfrac{625^{\left(\frac{3x}{4} - 1\right)} + 125^{(x - 1)}}{5^{(3x - 2)}}$ .
How do you simplify the top?
Answer
Write every number as a power of 5, then take out the smaller power: $5^{3x - 4} + 5^{3x - 3} = 5^{3x - 4}(1 + 5)$ .
From the lesson: Indices and exponential equations Try the question