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**Change of Base Formula**

*"Make a list of 5 real world situations where we would use a base other than Base 10 (log) and base e (ln)."*

*"So how do we do problems where we need to use these other bases?"*

**Take the log of the problem divided by the log of the base.**

log_{a} x = ( log_{b} x ) / ( log_{b} a )

_{a}x = ( log x ) / ( log a ) = ( ln x ) / ( ln a )

_{5}8 = ( ln 8 ) / ( ln 5 )

**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**

**The exponent on the argument is the multiplied by the log. (You could also say it becomes the coefficient of the log)**

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 )

**Do you have a Sub on Log Day? Here you go...**

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