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Exponentials

An exponential is a power , otherwise known as an index

A base number is raised to a power, the exponent.

Example

7³ has base 7, exponent 3

7³ means 7x7x7 , which has value 343

The inverse of an exponential function is called the logarithmic function

If y = a^x
then x = log_a y
To solve an exponential, take logs of both sides to the same base.

Scientific calculators usually have buttons for :-
Common logarithms natural logarithms
log for logarithms to base 10 ln for logarithms to base e =2.71828…

Example

in1

also

in2

ln3

The natural base: e

The natural base, e , uses Euler's number e = 2.71828182…

y = e^x has the special property that

, which is useful for differentiation at Advanced Higher

The inverse of y = e^x is x = log_ey, written x = ln y

Graphing Exponential and Logarithmic functions Excel -Exponential / log functions - graphs

Growth functions

A growth function is one where the output increases rapidly.

Example

£100 is deposited in a bank at a fixed rate of 12% per annum.
If A(n) = the amount of money in the account after n years,
a) show that A(n) = 100 x 1.12n
b) calculate the amount in the account after 10 years.

{This is the background behind CR^y }

The account growth looks like this:

Example

A factory has a target of 1.5% increase in output per year.
In 2003, the production was 18000 units.
In 2005, the production was 18515 units.
Was the production target met?

Decay functions

A decay function is one where the output decreases rapidly.

Example

boat

8000 gallons of oil are lost in an oil spill.
The clean up crew manage to clean 67 % of the oil each week.
a) How much oil is left after 1 week ?
b) How many weeks of cleaning are needed for there to be 10 gallons left ?

The oil spill decay looks like this:

Half life

The rate of decay of a radioactive source can be represented by the equation

where N is the number of radioactive atoms present at time t , λ is the transformation decay constant and N_o is the original starting value.

The half-life of carbon-14 is 5,730 ± 40 years and is used for radio carbon dating

Example

Given a half life of 5730 years, calculate the decay constant λ.

giving

Excel Spreadsheet

Log - Linear Graphs

Log scale Y axis only

If you have a graph with a logarithmic y axis, but ordinary x axis
then a straight line log y = (log b)x + log a
confirms a relationship of the form y = ab^x
for suitable constants a and b.

If y = ab^x then log y = log a + xlogb

This is because of the log laws

Compare this to Y = mx + c
where Y = log y, m = log b and c = log a

Example

Find the equation of the graph below in the form y = ab^x

loglin

The points (0,2) , (2,4) , (6,8) and (10, 12) lie on this graph,

giving c = 2 and m = 1

002

Now,

and

Likewise,

Putting all together, a =4, b = 2 so writing in the form y=ab^x the graph is y = 4(2^x)

This is not the same as y = 8^x , just as 2 x 4² is not equal to 8²

Notice :

When x = 0

When x = 2

When x = 6

When x = 10

Giving points (0,4) , ( 2, 16) ,( 6,256 ) and ( 10, 4096 )

Comparing with y = 2^x , it can be clearly seen that the graph has been scaled by a factor of 4 in the y direction.

comp

Example

Show that the formula connecting the following data is of the form y = ab^x .
Find the value of a and b and state the formula that connects x and y.

Solution

To show that y and x are related by the formula y=ab^x

Take logs to base 10 of y.
Plot against x.

A straight line confirms that a relationship exists.

Taking logs to base 10 of y gives

Plotting logy against x gives

Now take two points on the line of best fit to work out the gradient and y-intercept.
Use Y = mX + c, with Y = log y, m = logb , c=loga
Solve as simultaneous equations