Indices and Logarithms - fods12.files.wordpress.com

1

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


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


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


Note that here the equation was converted into a polynomial equation by substituting ,

which was then solved using the Null Factor Law

Example 7


Example 8

Prove that the following equation is true:

4

Example 9


Example 10


Example 11


5

Example 12


Example 13


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


Example 2


Note that all terms are written to the same number (3 in this case), and then simplified

8

Example 3


Note that the constant was converted to a logarithm, and then log laws were applied

Example 4


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


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


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


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


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


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


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

Indices and Logarithms - fods12.files.wordpress.com

Documents

Transcript of Indices and Logarithms - fods12.files.wordpress.com