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Laws of Logarithms (OCR AS Maths: Pure)
Revision Note
Laws of Logarithms
What are the laws of logarithms?
- There are many laws or rules of indices, for example
- am x an = am+n
- (am)n = amn
- There are equivalent laws of logarithms (for a > 0)
- There are also some particular results these lead to
- Two of these were seen in the notes Logarithmic Functions
- Beware …
- … log (x + y) ≠ log x + log y
- Results apply to ln too
- In particular and
How do I use the laws of logarithms?
- Laws of logarithms can be used to …
- … simplify expressions
- … solve logarithmic equations
- … solve exponential equations
Examiner Tip
- Remember to check whether your solutions are valid
- log (x+k) is only defined if x > -k
- You will lose marks if you forget to reject invalid solutions
Worked example
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"ln"
What is ln?
- ln is a function that stands for natural logarithm
- It is a logarithm where the base is the constant "e"
- It is important to remember that ln is a function and not a number
What are the properties of ln?
- Using the definition of a logarithm you can see
- is only defined for positive x
- As ln is a logarithm you can use the laws of logarithms
How can I solve equations involving e & ln?
- The functions and are inverses of each other
- If then
- If then
- If your equation involves "e" then try to get all the "e" terms on one side
- If "e" terms are multiplied, you can add the powers
- You can then apply ln to both sides of the equation
- If "e" terms are added, try transforming the equation with a substitution
- For example: If then
- You can then solve the resulting equation (usually a quadratic)
- Once you solve for y then solve for x using the substitution formula
- If "e" terms are multiplied, you can add the powers
- If your equation involves "ln", try to combine all "ln" terms together
- Use the laws of logarithms to combine terms into a single term
- If you have then solve
- If you have then solve
Worked example
Examiner Tip
- Always simplify your answer if you can
- for example,
- you wouldn't leave your final answer as so don't leave your final answer as
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