Define exponential growth .

Exponential growth is a quantity increasing from a starting amount by the same scale factor in each equal time interval. That interval need not be a year: it can be a day, an hour or a minute, as long as it is the same one each time.

IGCSEMathsCambridge (CIE)International MathsExtendedFlashcardsAlgebra & SequencesExponentials & Logarithms

Exponentials & Logarithms (Cambridge (CIE) IGCSE International Maths: Extended): Flashcards

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FrontExponential Growth & Decay
Complete the rule for the model $y = a^{x}$ used in growth and decay problems.
A value of a greater than 1 gives exponential $______$ while a value between 0 and 1 gives exponential $______$ instead.

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Complete the rule for the model $y = a^{x}$ used in growth and decay problems.
A value of a greater than 1 gives exponential $______$ while a value between 0 and 1 gives exponential $______$ instead.
The completed rule is:
A value of a greater than 1 gives exponential growth while a value between 0 and 1 gives exponential decay instead.
The scale factor a is applied once in every equal time interval.
In the model $y = b a^{x}$ what does the value of b represent?
b is the initial amount of the quantity being modelled.
When $x = 0$ the power term equals 1, so $y = b$ at the very start of the model.
A bird population is modelled by $N = 1 . 04^{t}$ where N is measured in thousands. What is the population after 13 years?
Substituting gives $1 . 04^{13} = 1.66507 \dots$ thousand birds.
Multiplying by 1000 gives about 1665, which is 1700 birds to the nearest hundred.
True or False?
A model of the form $y = a^{x}$ shows decay when a is negative.
False.
The base a can never be negative in this model at all.
A negative base would make y flip between positive and negative as x moved from one whole number to the next, which no real quantity does.
A model $y = a^{x}$ gives $y = 0.512$ when x is 3. What is the value of a?
Substituting gives $0.512 = a^{3}$ as the equation to solve.
Taking the cube root of both sides gives $a = 0.8$ for this model.
A model $y = 0 . 8^{x}$ gives the fraction of concert tickets still unsold. Why might it not be realistic?
The value of y gets closer and closer to zero but never actually reaches it.
The model therefore says the tickets never completely sell out, which cannot happen in practice.
Define exponential growth.
Exponential growth is a quantity increasing from a starting amount by the same scale factor in each equal time interval.
That interval need not be a year: it can be a day, an hour or a minute, as long as it is the same one each time.
Define a logarithm.
A logarithm is the inverse of an exponential: $\log_{a} (b)$ is the power you raise a to in order to get b.
So $a^{x} = b$ and $x = \log_{a} (b)$ say exactly the same thing in two different ways.
What is the value of $\log_{5} 125$ and why?
It is 3, because 3 is the power you raise 5 to in order to reach 125.
In other words $5^{3} = 125$ gives the answer directly.
Complete the sentence about the base of a logarithm.
When a logarithm is written with no base at all, such as $\log (x)$ on its own, the base is taken to be $______$ by convention.
The completed sentence is:
When a logarithm is written with no base at all, such as $\log (x)$ on its own, the base is taken to be 10 by convention.
The plain log button on a calculator always uses this base.
True or False?
$\log_{5} 1$ is equal to 0.
True.
The power you raise 5 to in order to get 1 is zero, since $5^{0} = 1$ always holds.
The same happens whatever the base, so a logarithm of 1 is always 0.
Can a logarithm be negative, or a fraction? Give an example of each.
Yes to both, because the power itself can be negative or fractional.
For instance $\log_{5} (\frac{1}{5}) = - 1$ and $\log_{5} (\sqrt{5}) = \frac{1}{2}$ show each case.
How do you solve $2^{x} = 10$ using logarithms?
Make x the subject to get $x = \log_{2} 10$ and evaluate that on a calculator.
Working in base 10 throughout gives the same value, since $x = \frac{\log 10}{\log 2} = 3.3219 \dots$ as well.
Solve $\log (8 x) = 4$ without a calculator, where the base is 10.
Rewrite it as an exponential statement, so $10^{4} = 8 x$ follows directly.
That gives $8 x = 10000$ and therefore $x = 1250$ as the solution.
An amount of 500 is invested at 5% per year. After how many whole years does it first exceed 900?
Solve $500 \times 1 . 05^{n} = 900$ by making n the subject with logarithms.
That gives $n = \log_{1.05} 1.8 = 12.05 \dots$ so the amount first exceeds 900 after 13 whole years.

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