5.4 – Properties and Applications of Logarithims

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

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

Example. Yo! Decompose this mess! ln( )

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

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

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)

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

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

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

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 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?

Assignment Pg odd, odd, 85, 86, 92,

Solutions 25) 1/531) 9 34) -2 36) No Solution 40) ) ) 10 60) 4 = log ) e x = log ) 81 = ) W = ) e 3 = 5x