1. 5.4 – Properties and Applications of Logarithims

2. Three properties of logarithms correspond to properties of exponents 1) log a (xy) = log a (x) + log a (y) 2) log a (x/y) = log a (x) – log a (y) 3) log a (x r ) = r log a x These properties can be used to expand particular expressions

3. Example. Use the previous properties to expand the expression as much as humanly possible. log 4 (64x 3 y 3 )

4. Example. Yo! Decompose this mess! ln( )

5. We can also use the properties to condense an expression. Example. Condense the following expression. ln(x 2 ) – ½ ln(y) + ln (2)

6. Example. Condense the following expression. 3log 7 2 – 2log 7 4

7. Change of Base Recall… With our calculators, we can calculate logs; but only in base 10 To overcome this issue, we can use what is known as the change of base Log b x = log a x/log a b OR ln(x)/ln(b)

8. Example. Evaluate the following: A) log 7 15 B) log 0.2 17 C) log 1/5 625

9. Applications Example. The pH of a solution is defined as –log[H 3 O + ], where [H 3 O + ] is the concentration of hydronium ions in moles/liter. A pH less than 7 is said to be acidic. Greater than 7 is said to be basic

10. Example. A carton of orange juice is found to have a [H 3 O + ] concentration of 1.58 x 10 -4 moles/liter. What is the pH? How can we use our equation?

11. Example. A person measures the pH in their pool using a basic kit. The person finds the [H 3 O + ] to be 2.40 x 10 -8 moles per liter. It’s said to be safe if the pH is between 7.2 and 7.6. Is it safe to swim in their pool?

12. Assignment Pg. 425 5-11 odd, 19-27 odd, 85, 86, 92,

13. Solutions 25) 1/531) 9 34) -2 36) No Solution 40) 41.9644) 5.6 49) 10 60) 4 = log 5 625 70) e x = log 2 1172) 81 = 3 4 78) W = 5 12 83) e 3 = 5x