A LevelMathsEdexcelPureFlashcardsExponentials & LogarithmsExponential & Logarithms

Exponential & Logarithms (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

1/22

0Still learning

Know0

FrontExponential Functions
Define exponential function.

Was this flashcard helpful?

Cards in this collection (22)

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.
$a^{x} = b$ and $\log_{a} b =______$ are equivalent statements, in which $a$ is called the $______$ .
$a^{x} = b$ and $\log_{a} b = x$ are equivalent statements, in which $a$ is called the base.
The base must be positive, and the logarithm is simply the power.
How do you read $\log_{3} 81 = 4$ in words?
"The power you raise $3$ to, to get $81$ , is $4$ ."
Saying a logarithm statement aloud like that is the quickest way to check it is the right way round.
How do you evaluate $\log_{2} 32$ without a calculator?
Ask what power of $2$ gives $32$ : since $2^{5} = 32$ , the answer is $5$ .
Knowing the first few powers of $2$ , $3$ , $4$ , $5$ and $10$ makes most such logarithms immediate.
What does $\log x$ mean when no base is written?
Base $10$ : $\log x$ means $\log_{10} x$ , and it is sometimes written $\lg x$ instead.
A base written explicitly, such as $\log_{2}$ , always overrides that convention.
True or False?
${(\log x)}^{2}$ and $\log x^{2}$ mean the same thing.
False.
${(\log x)}^{2}$ means take the logarithm first and then square it.
$\log x^{2}$ means square $x$ first and then take the logarithm, which the power law turns into $2 \log x$ .
What does it mean to say a logarithm is the inverse of raising to a power?
Each undoes the other, so $a^{\log_{a} x} = x$ and $\log_{a} (a^{x}) = x$ .
That is what makes a logarithm the tool for getting a variable down out of an exponent.
Define e.
e is an irrational number, approximately $2.718$ , sometimes called Euler's number.
Like $π$ , it is a number, not a variable and not a function.
What is special about the graph of $y = e^{x}$ ?
Its gradient at every point is equal to its own value, so $\frac{d y}{d x} = e^{x}$ .
No other base behaves that way, which is exactly why $e$ is singled out.
Like every exponential graph, $y = e^{x}$ passes through $(0,______)$ and has the $______$ -axis as an asymptote.
Like every exponential graph, $y = e^{x}$ passes through $(0, 1)$ and has the $x$ -axis as an asymptote.
Since $e \approx 2.718$ is greater than $1$ , the curve is a growth curve.
How is the graph of $y = e^{- x}$ related to $y = e^{x}$ ?
It is the reflection in the $y$ -axis.
The two are $f (x)$ and $f (- x)$ , which is precisely what that reflection produces.
True or False?
$e$ can be written exactly as a fraction.
False.
$e$ is irrational, so it has no exact fractional form, exactly like $π$ .
$2.718$ is only an approximation, which is why answers are often left in terms of $e$ .
If $y = e^{k x}$ then $\frac{d y}{d x} =______$ , and if $y = e^{- k x}$ then $\frac{d y}{d x} =______$ .
If $y = e^{k x}$ then $\frac{d y}{d x} = k e^{k x}$ , and if $y = e^{- k x}$ then $\frac{d y}{d x} = - k e^{- k x}$ .
The constant from the power comes down as a multiplier, and $y = e^{x}$ is simply the case $k = 1$ .
What stays the same when you differentiate $e^{k x}$ ?
The exponential part itself: $e^{k x}$ reappears in the derivative unchanged.
Only a constant multiplier is added in front, which is what makes exponential derivatives unusually simple.
What is the derivative of $y = e^{- 3 x}$ ?
$\frac{d y}{d x} = - 3 e^{- 3 x}$ .
The minus sign comes down with the $3$ , so a decay curve has a negative gradient everywhere along it.
True or False?
The gradient of $y = e^{x}$ is never zero.
True.
The gradient equals $e^{x}$ , which is positive for every value of $x$ .
So the curve is always increasing and has no stationary points at all.
How do you find the gradient of $y = e^{2 x}$ at $x = 0$ ?
Differentiate to get $\frac{d y}{d x} = 2 e^{2 x}$ , then substitute $x = 0$ .
Since $e^{0} = 1$ , the gradient there is $2$ .

Sign up to unlock flashcards

By signing up you agree to our Terms and Privacy Policy

Previous:Modelling with Trigonometric FunctionsNext:Laws of Logarithms