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Change of Base Formula
log_{a} x = ( log_{b} x ) / ( log_{b} a )
Multiplication Rule
The log of a product is the sum of the logs.
log_{a} xy = log_{a} x + log_{a} y
Division Rule
The log of a quotient is the difference of the logs.
log_{a} (x/y) = log_{a} x – log_{a} y
Raising to a Power Rule
log_{a} x^{r} = r * log_{a} x
Summary – Properties of Logarithms
 The log of a product is the sum of the logs
 The sum of the logs is the log of the products
 The log of a quotient is the difference of the logs
 The difference of the logs is the log of the quotient
 The exponent on the argument is the coefficient of the log
 The coefficient of the log is the exponent on the argument
Most Common Mistakes
 The log of a sum is NOT the sum of the logs. The sum of the logs is the log of the product. The log of a sum cannot be simplified.
log_{a} (x + y) ≠ log_{a} x + log_{a} y
 The log of a difference is NOT the difference of the logs. The difference of the logs is the log of the quotient. The log of a difference cannot be simplified.
log_{a} (x – y) ≠ log_{a} x – log_{a} y
 An exponent on the log is NOT the coefficient of the log. Only when the argument is raised to a power can the exponent be turned into the coefficient. When the entire logarithm is raised to a power, then it can not be simplified.
(log_{a} x)^{r} ≠ r * log_{a} x
 The log of a quotient is not the quotient of the logs. The quotient of the logs is from the change of base formula. The log of a quotient is the difference of the logs.
log_{a} (x / y) ≠ ( log_{a} x ) / ( log_{a} y )
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