The Laws of Logs (Leaving Cert Mathematics): Revision Notes
The Laws of Logs
Logarithms are the inverse of exponentials. If , then . This means logarithms answer the question: "To what power must the base be raised to get ?"
Recap:
- Base: The number is the base of the logarithm, and it must be and .
- Argument: The value inside the logarithm () is called the argument and must satisfy .
The following are the key laws of logarithms, which help simplify expressions and solve equations involving logs:
- Product Rule:
This means the logarithm of a product equals the sum of the logarithms of the factors.
- Quotient Rule:
The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator.
- Power Rule:
If the argument has an exponent, you can bring the exponent in front as a multiplier.
- Negative Argument Rule:
The logarithm of the reciprocal of a number equals the negative logarithm of the number.
- Logarithm of 1:
This is because for any base .
- Logarithm Identity:
Raising the base aa to the logarithm of returns .
- Inverse Property:
The logarithm of a base raised to a power gives the power itself.
Example
Evaluate the .
Summary:
- Logarithms are the inverse of exponents.
- These laws are crucial for solving logarithmic and exponential equations.