Lesson 3-‐ The Laws of Logarithms Warm up problem:
The blue graph cab be generated by stretching the red graph of 𝑦 = 𝑙𝑜𝑔! 𝑥. Write the equation that describes the blue graph.
We are familiar with exponent laws (i.e. the product law, the power law, etc.) and evaluating powers with the same base. Since a logarithm is an exponent, the laws of exponents can be used to develop a set of laws for logarithm.
Logarithmic Laws Product law for logarithms: log! 𝑥 + log! 𝑦 = log!(𝑥𝑦) where x , and y are positive real numbers.
Evaluate or simplify the following logarithmic expressions using the product law for logarithms Ex1: log!(128 ×64) Ex2: log! 25𝑎 Ex3: log! 4 + log! 9
Ex4: log 𝑥 + 1 + (𝑥 − 3) Ex5: Express log! 7 + log! 9 + log! 3𝑛 as a single simplified logarithm
Ex6: log! 7 7
This law states that the sum of two logarithms with the same base b, is equal to the logarithm of the product of those two numbers.
Quotient law for logarithms: log! 𝑥 − log! 𝑦 = log!( ! ! ) where x , and y are positive real numbers
Evaluate or simplify the following logarithmic expressions using the quotient law for logarithms Ex1: log!(
!! )
Ex2: If log!𝑚 = 𝑥, determine an expression in terms of x that’s equivalent to log!(
!"!)
Ex3: log! 245 − log! 5 Ex4: Express log!(72𝑥!)−log!(9𝑥) as a simplified logarithm
Power law for logarithms: log! 𝑥! = 𝑛 log! 𝑥 ; where x , and y are positive real numbers Evaluate or simplify the following logarithmic expressions using the power law for logarithms Ex1: log!( 67! ) Ex2: Determine an expression written in terms of k that is equivalent to log!(243!!!)
This law states that the difference of two logarithms with the same base b, is equal to the logarithm of the quotient of those two numbers.
This law states that the logarithm of a power 𝑥!, is equal to the product of the exponent of that power, n, and the logarithm of the base of that power, x, for any base, b. The exponent becomes the multiplier.