5.4 – Properties and Applications of Logarithims

• Three properties of logarithms correspond to properties of exponents

• 1) loga(xy) = loga(x) + loga(y)

• 2) loga(x/y) = loga(x) – loga(y)

• 3) loga(xr) = r logax

• These properties can be used to expand particular expressions

• Example. Use the previous properties to expand the expression as much as humanly possible.

• log4(64x3y3)

• Example. Yo! Decompose this mess!

• ln( )3

2

q

pe

• We can also use the properties to condense an expression.

• Example. Condense the following expression.

• ln(x2) – ½ ln(y) + ln (2)

• Example. Condense the following expression.

• 3log72 – 2log74

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

• Logbx = logax/logab OR ln(x)/ln(b)

• Example. Evaluate the following:

• A) log715

• B) log0.217

• C) log1/5625

Applications

• Example. The pH of a solution is defined as –log[H3O+], where [H3O+] 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 [H3O+] concentration of 1.58 x 10-4 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 [H3O+] 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?

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

• Solutions25) 1/5 31) 9 34) -2 36) No Solution40) 41.96 44) 5.6 49) 10 60) 4 = log5625

70) ex = log211 72) 81 = 34 78) W = 512

83) e3 = 5x