6.1.2
Logarithms
Logarithms
Logarithms
In order to solve equations with exponential functions in them, we need to define the inverse of the exponential operation.
![Illustrative background for Logarithm](https://image-v2.cdn.app.senecalearning.com/2020-01/aff2168f-fd67-4519-9fdc-818bc90db8f2/Screenshot%202020-01-24%20at%2015.30.49-min,h_400,q_80,w_640.png)
![Illustrative background for Logarithm ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/aff2168f-fd67-4519-9fdc-818bc90db8f2/Screenshot%202020-01-24%20at%2015.30.49-min,h_400,q_80,w_640.png)
Logarithm
Logarithm
- The logarithmic function is written as , which is equivalent to
.
- is called the base of the logarithm.
- To remember this, think "left to the right equals middle".
- The base 2 logarithm of the number 8 is 3, so
![Illustrative background for Range and domain](https://image-v2.cdn.app.senecalearning.com/2020-01/aff2168f-fd67-4519-9fdc-818bc90db8f2/Screenshot%202020-01-24%20at%2015.30.49-min,h_400,q_80,w_640.png)
![Illustrative background for Range and domain ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/aff2168f-fd67-4519-9fdc-818bc90db8f2/Screenshot%202020-01-24%20at%2015.30.49-min,h_400,q_80,w_640.png)
Range and domain
Range and domain
- The range of the exponential function is the set of positive real numbers.
- This means the domain of a logarithm is the set of positive real numbers.
- So we can only define for .
- We also specified that the allowed range of values of the base of the exponential function was the positive real numbers excluding the number 1.
- So we can only define for .
![Illustrative background for Properties](https://image-v2.cdn.app.senecalearning.com/2020-01/74529e9d-4948-4564-89a1-2be665b0acb8/Screenshot%202020-01-24%20at%2015.35.35-min,h_400,q_80,w_640.png)
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Properties
Properties
- For single variables and numbers, we often drop the parenthesis to make logarithms easy to write.
- The logarithm of the number one is equal to zero for any base, so:
- For logarithms of base 10, we often drop the base number and write:
![Illustrative background for Graph](https://image-v2.cdn.app.senecalearning.com/2020-01/62d434c9-b9dc-463e-8674-c18c93ac9850/Screenshot%202020-01-24%20at%2015.30.56-min,h_400,q_80,w_640.png)
![Illustrative background for Graph ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/62d434c9-b9dc-463e-8674-c18c93ac9850/Screenshot%202020-01-24%20at%2015.30.56-min,h_400,q_80,w_640.png)
Graph
Graph
- The graph of does not cross the -axis.
- The -intercept is 1.
- The orientation of the graph is different for values of and as shown.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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