Laws of Logarithms
What are the laws of logarithms?
- Laws of logarithms allow you to simplify and manipulate expressions involving logarithms
- They can help with solving exponential and logarithmic equations
- The laws of logarithms are closely related to the laws of indices
- You need to know the following laws, which are valid for , :
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- "log of a product is equal to the sum of the logs"
- This relates to
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- "log of a division is equal to the difference of the logs"
- This relates to
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- "power in a log may be brought down as a multiplier in front of the log"
- note that
- This relates to
-
- This relates to
-
- This relates to
-
- Be careful
- With the first two laws the logs on the right-hand side must have the same base
- Logs with different bases cannot be combined using those laws
- Also note the following (students often make these mistakes on the exam):
- is not equal to
- is not equal to
- With the first two laws the logs on the right-hand side must have the same base
What are some other useful properties of logarithms?
- You should also be familiar with the following properties of logarithms
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- This can be derived from the third and fourth laws above
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- Because
- Then use the third law above
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- Because and are inverse functions
- "log cancels exponential and exponential cancels log"
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- Also remember that is another way of writing
- All the laws and properties apply to as well
- This includes
Exam Tip
- Make sure you know the laws and properties of logarithms
- They are not included on the exam formula sheet
Worked example
Substitute that, and your answer from part (a), into the equation and solve for