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Indices and Logarithms
Indicial Laws
Negative numbers cannot be expressed in index form
Indicial equations can be solved by substituting for
Classic one payment per period growth is modelled by
Continuous growth is modelled by where is the initial value and is the
interest rate expressed as a decimal (e.g. )
General exponential growth formula:
, where is the number of
payments per period, is the number of periods
Example 1
Simplify
Example 2
Simplify the following expression:
2
Note that the method here was to multiply both fractions by their indices, then combine the
fractions, and then simplify.
Example 3
Simplify the following expression:
Note that the method here was to write each number as the product of two or more prime
numbers, then remove the brackets and individually apply all the indices, then combine like terms
and simplify
Example 4
Simplify the following expression:
3
Note that the method here was to convert negative indices to positive indices, consolidate the
fractions into two, and then cross multiply and simplify
Example 5
Solve the following equation to find :
Example 6
Solve the following equation to find :
Note that here the equation was converted into a polynomial equation by substituting ,
which was then solved using the Null Factor Law
Example 7
Solve the following equation to find :
Example 8
Prove that the following equation is true:
4
Example 9
Simplify the following expression:
Example 10
Simplify the following expression:
Example 11
Solve the following equation to find :
5
Example 12
Solve the following equation to find :
Example 13
Simplify the following expression:
Note that here we had to take out a common factor from both the numerator and the denominator
such that the remaining portions were the same, and could be cancelled out
6
Exponential Functions Exponential functions are functions of the form , where is any positive real number
other than 1
The basic formula is
The higher the value of , the steeper the graph
is the dilation factor in the y-direction
is the dilation factor in the x-direction
The graph is translated units to the left, or – to the right
The graph is translated units up, or – down
Y-intercept occurs at
X-intercepts occur only if
Horizontal asymptote occurs at
Transformation Formula Graph
Dilation in y-direction: graph is pushed times further away from x-axis
Dilation in x-direction:
graph is pushed
times
further away from y-axis
Horizontal translation
Vertical translation
7
Reflection in the x-axis
Reflection in the y-axis
Logarithm Laws The logarithm of a number to a given base is the power to which the base must be raised in order to
produce the number. Logarithms answer the question ‘what power do I need to raise to in order
to get ?’
If
The base of a logarithm must be a real positive number greater than 1
Example 1
Simplify the following expression:
Example 2
Simplify the following expression:
Note that all terms are written to the same number (3 in this case), and then simplified
8
Example 3
Simplify the following expression:
Note that the constant was converted to a logarithm, and then log laws were applied
Example 4
Simplify the following expression:
Here the numbers were combined using log laws, and then equalised with the base so as to
eliminate the logarithm
Example 5
Find the value of the following expression:
As most calculators cannot calculate bases of numbers other than 10 or , it is necessary to convert
other bases using the change of bases rule
Example 6
Solve the following equation to find :
When the unknown to be evaluated is the number in the logarithm, convert the equation from
logarithmic to indicial form and solve
9
Example 7
Solve the following equation to find :
When the unknown to be evaluated is the base, convert the equation from logarithmic to indicial
form and solve using whatever method is applicable (in this case equating the exponents, in other
cases the quadratic formula)
Example 7
Solve the following equation to find :
When the unknown to be evaluated is the answer, convert the equation from logarithmic to indicial
form and solve using whatever method is applicable
Example 8
Solve the following equation to find :
Here the logarithm terms were combined, and then the logarithms were dropped to form a
quadratic equation, which was solved using the Null Factor Law
Example 9
Solve the following equation to find :
10
Here the logarithm was converted to indicial form, before both sides were raised to the correct
power that would isolate .
Example 10
Find the value of the following expression:
Example 11
Solve the following equation to find :
11
Logarithmic Functions Exponential functions are functions of the form , where is any positive real number
other than 1
The basic formula is
The higher the value of , the shallower the graph
is the dilation factor in the y-direction (from the x-axis)
is the dilation factor in the x-direction (from the y-axis)
The graph is translated units to the left, or – to the right
The graph is translated units up, or – down
Y-intercept occurs at
X-intercept occurs at
if
Vertical asymptote occurs at
Transformation Formula Graph
Dilation in y-direction: graph is pushed times further away from x-axis
Dilation in x-direction:
graph is pushed
times
further away from y-axis
Horizontal translation
Vertical translation
12
Reflection in the x-axis
Reflection in the y-axis