Warm Up – 4.28 - Monday 1. Write each logarithm as an exponential: log 3 10 =𝑥−1 B) ln 𝑦=5𝑥 2. Write each exponential as a logarithm. 4 𝑥 =15 B) 𝑒 𝑥−3 =8 3. Solve the logarithmic equation. log 5 𝑥+1 = log 5 12 log 2 3𝑦+1 = log 2 (2𝑦)

Rational Exponents When I raise an exponent to a fraction, that is the same as taking the root of the denominator. Example: 4 5/3 = 3 4 5 Example: 7 −4/5 = 1 5 7 4

Rational Exponents Classwork

Rules of Logarithms log 𝑎 𝑏 + log 𝑎 𝑐 = log 𝑎 (𝑏∙𝑐) THESE RULES ONLY APPLY IF THE BASE IS THE SAME!!! log 𝑎 𝑏 + log 𝑎 𝑐 = log 𝑎 (𝑏∙𝑐) log 𝑎 𝑏 − log 𝑎 𝑐 = log 𝑎 𝑏 𝑐 log 𝑎 𝑏 𝑐 =𝑐∙ log 𝑎 𝑏

Expanding Logarithms These problems are finished when each logarithm has only one number or integer on the inside. Example: log 2 𝑥 + log 2 ⁡(𝑦∙𝑧) The first log is as simplified as it gets. The second log has two variables and needs to be expanded.

Example #1 log 2 𝑥 + log 2 ⁡(𝑦∙𝑧)

Example #1 - Solution log 2 𝑥 + log 2 ⁡(𝑦∙𝑧) When we multiply on the inside we add the logs. log 2 𝑥 +( log 2 𝑦 + log 2 (𝑧)) Notice the parentheses. These are not necessary here because there is a positive in front of them. log 2 𝑥 + log 2 𝑦 + log 2 𝑧

Example #2 log 2 𝑥𝑦 𝑧

Example #2 - Solution log 2 𝑥𝑦 𝑧 Take care of the division before the multiplication. log 2 𝑥𝑦 − log 2 𝑧 (log 2 𝑥 + log 2 𝑦 )− log 2 𝑧 We can drop the parenthesis because there is no negative in front. log 2 𝑥 + log 2 𝑦 − log 2 𝑧

Example #3 log 4 𝑥 𝑦𝑧

Example #3 - Solution log 4 𝑥 𝑦𝑧 log 4 𝑥 − log 4 𝑦𝑧 There is a minus sign in front of the parenthesis so we change the signs on the inside. log 4 𝑥 − log 4 𝑦 − log 4 (𝑧)

Example #4 log 3 2 9 3

Example #4 - Solution log 3 2 9 3 When everything is raised to an exponent, deal with the exponent first. 3 log 3 2 9 3( log 3 2 − log 3 (9)) Distribute the 3. 3 log 3 2 −3 log 3 9

Example #5 log 5 (𝑥∙ 𝑦 7 )

Example #5 - Solution log 5 (𝑥∙ 𝑦 7 ) When only one base is raised to an exponent, deal with the multiplication first! log 5 𝑥 + log 5 ( 𝑦 7 ) log 5 𝑥 +7 log 5 𝑦