Laws of Logarithms (Edexcel International A Level (IAL) Maths: Pure 2): Flashcards

Exam code: YMA01

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Define exponential function.
A function of the form $y = a^{x}$ , where the variable sits in the power and $a > 0$ .
It is the variable's position that makes it exponential, not the size of the base.
What do all graphs of the form $y = a^{x}$ have in common?
They all pass through $(0, 1)$ , because $a^{0} = 1$ whatever $a$ is.
The $x$ -axis is an asymptote, so the curve approaches it without ever reaching it.
$y = a^{x}$ shows exponential growth when $a______$ , and exponential decay when $______< a <______$ .
It shows growth when $a > 1$ , and decay when $0 < a < 1$ .
A base bigger than $1$ multiplies up at every step, while a base between $0$ and $1$ multiplies down.
True or False?
$y = 1^{x}$ is an exponential function.
False.
$1^{x} = 1$ for every value of $x$ , so the graph is just the horizontal line $y = 1$ .
That is precisely why $a = 1$ is excluded: nothing grows and nothing decays.
For $x > 0$ , is $3^{x}$ above or below $2^{x}$ , and what happens for $x < 0$ ?
Above for $x > 0$ , and below for $x < 0$ .
The two curves meet at $(0, 1)$ , which every exponential graph passes through, so that is where the ordering swaps.
Why must the base of an exponential function be positive?
Because a negative base gives no real value for fractional powers.
${(- 4)}^{\frac{1}{2}}$ does not exist as a real number, so the graph would have gaps instead of being a smooth curve.

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