KS4 Grade 9 - INDICES, SURDS AND LOGARITHMS
Lesson 5...LOGARITHMS (COMMON AND NATURAL)
There are two special types of Logarithm as below.
COMMON LOGARITHMS are Logarithms to the base 10. Common Logarithms are often denoted by the abbreviation 'lg'
NATURAL LOGARITHMS are Logarithms to the base e. Natural Logarithms are often denoted by the abbreviation 'ln'
Common and Natural Logarithms can be solved using the appropriate function buttons on a scientific calculator.
Eg. e^3x = 9
ln(9) = 3x
x = ln(9) / 3
x = 0.732 (3s.f.)
TASK 1 - Textbook Pg47 Ex3.4
HOMEWORK - Holiday HMWK WS
COMMON LOGARITHMS are Logarithms to the base 10. Common Logarithms are often denoted by the abbreviation 'lg'
NATURAL LOGARITHMS are Logarithms to the base e. Natural Logarithms are often denoted by the abbreviation 'ln'
Common and Natural Logarithms can be solved using the appropriate function buttons on a scientific calculator.
Eg. e^3x = 9
ln(9) = 3x
x = ln(9) / 3
x = 0.732 (3s.f.)
TASK 1 - Textbook Pg47 Ex3.4
HOMEWORK - Holiday HMWK WS
g9add_hw-1.pdf | |
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Lesson 4...LOGARITHMS INTRO
In exponential equations where the index, as a variable, cannot be easily found, Logarithms can be used to determine its value, using the principle shown in the image on the left.
In the definition at the top on the left, if b (base) is raised to the power y (exponent), and the result equals x, then using Logarithms we can alter this as represented by the middle equation. This is read as 'the logarithm of x to base b' or 'log, base b, of x'
A scientific calculator may then be used to find the value of y using the 'log' function button.
Note: for 'log, base b, of x' above to be defined, then x > 0, a > 0, a ≠ 0
TASK 1 - Textbook Pg45 Ex3.3
HOMEWORK - Finish TASK 1
In the definition at the top on the left, if b (base) is raised to the power y (exponent), and the result equals x, then using Logarithms we can alter this as represented by the middle equation. This is read as 'the logarithm of x to base b' or 'log, base b, of x'
A scientific calculator may then be used to find the value of y using the 'log' function button.
Note: for 'log, base b, of x' above to be defined, then x > 0, a > 0, a ≠ 0
TASK 1 - Textbook Pg45 Ex3.3
HOMEWORK - Finish TASK 1
Lesson 3...EXPONENTIAL EQUATIONS
An equation that contains a variable in an index is called an indicial or EXPONENTIAL EQUATION
The simplest form is a^x = b. If b can be expressed as a^n, then a^x = a^n --> x = n, where a ≠ -1, 0, 1
To solve an exponential equation, such as that shown in the image on the left, firstly the bases must be changed so that they are the same. At this point the bases can be removed, leaving a solvable equation in x.
TASK 1 - Textbook Pg41 Ex3.32
HOMEWORK - EDX Simultaneous Equations worksheets
The simplest form is a^x = b. If b can be expressed as a^n, then a^x = a^n --> x = n, where a ≠ -1, 0, 1
To solve an exponential equation, such as that shown in the image on the left, firstly the bases must be changed so that they are the same. At this point the bases can be removed, leaving a solvable equation in x.
TASK 1 - Textbook Pg41 Ex3.32
HOMEWORK - EDX Simultaneous Equations worksheets
Lesson 2...SURDS AND RATIONALISING THE DENOMINATOR
If a root cannot be evaluated exactly (Eg. √5), for ease it can be left in this SURD form. All surds in this form are irrational numbers.
However, the product of two surds need not be a surd (Eg. √3 √3 = 3). This knowledge can be used to RATIONALISE THE DENOMINATOR of a fraction, where the denominator is a surd. The product of (a√h + b√k) and (a√h - b√k) is a rational number and this allows us to simplify fractions as in the example in the image on the left, by multiplying both the numerator and denominator by the expression 2 + √3
TASK 1 - Textbook Pg39 Ex3.1 Q7
TASK 2 - Rationalising the Denominator worksheet
HOMEWORK - None set
However, the product of two surds need not be a surd (Eg. √3 √3 = 3). This knowledge can be used to RATIONALISE THE DENOMINATOR of a fraction, where the denominator is a surd. The product of (a√h + b√k) and (a√h - b√k) is a rational number and this allows us to simplify fractions as in the example in the image on the left, by multiplying both the numerator and denominator by the expression 2 + √3
TASK 1 - Textbook Pg39 Ex3.1 Q7
TASK 2 - Rationalising the Denominator worksheet
HOMEWORK - None set
Lesson 1...INDICES (EXPONENTS)
When n (index, exponent, power) is a positive integer and a is the base, a^n is defined as:
a^n = a x a x a x ... x a [where the number of a values is equal to n]
There are a number of rules governing calculations involving indices. These can be see in the image on the left.
Eg. 9^(1/3) x 9^(1/6) = 9^(2/6) x 9^(1/6) = 9^(3/6) = 9^(1/2) = √9 = 3
TASK 1 - Textbook Pg39 Ex3.1 Q1-6
HOMEWORK - None set
a^n = a x a x a x ... x a [where the number of a values is equal to n]
There are a number of rules governing calculations involving indices. These can be see in the image on the left.
Eg. 9^(1/3) x 9^(1/6) = 9^(2/6) x 9^(1/6) = 9^(3/6) = 9^(1/2) = √9 = 3
TASK 1 - Textbook Pg39 Ex3.1 Q1-6
HOMEWORK - None set