5.4 – Properties and Applications of Logarithims

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( )

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,

Solutions 25) 1/5 31) 9 34) -2 36) No Solution 40) 41.96 44) 5.6 49) 10 60) 4 = log5625 70) ex = log211 72) 81 = 34 78) W = 512 83) e3 = 5x

5.4 – Properties and Applications of Logarithims