14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms a have been applied correctly at each step.

1. Use properties of logarithms to evaluate, expand, or condense such expressions to be able to solve problems involving logarithms. 2. Solve exponential and logarithmic equations. 3. Rewrite equivalent logarithm expression by changing base (if we have time, else next week)

Example 1 Use Properties of Logarithms Use log and log to find the value of the expression to the nearest thousandth. ≈≈ a. log b. log 7 10 c. log 7 25 SOLUTION a. log = log 7 2 log 7 5 – Quotient property = – Simplify.Use the given values of log 7 2 and log – ≈

Example 1 Use Properties of Logarithms = Simplify. log 7 2 log 7 5 = + Product property ≈ + Use the given values of log 7 2 and log 7 5. c. log 7 25 = log Express 25 as a power. = 2 log 7 5 Power property 2 ≈ () Use the given value of log 7 5. = Simplify. b. log 7 10 = () 52 Express 10 as a product. log 7

Example 2 Expand a Logarithmic Expression Expand the expression. Assume all variables are positive. a. log 4 5x 2 log 7 y 3x3x b. SOLUTION a. log 4 5x 2 = log 4 5 log 4 x 2 + Product property = log log 4 x + Power property log 7 y 3x3x b. = log 7 3x log 7 y – Quotient property

Example 2 Expand a Logarithmic Expression = log 7 3 log 7 x log 7 y – Product property +

Example 3 Condense a Logarithmic Expression Condense the expression. a. log 16 2 log 2 – 3 log 5 log 4 b. + SOLUTION a. log 16 2 log 2 – = log 16 log 2 2 – Power property = log 2 16 Quotient property = log 4 Simplify.

Example 3 Condense a Logarithmic Expression 3 log 5 log 4 b. + = Power property log 5 3 log 4 + Product property = () log Simplify. = log 500

Checkpoint Use log and log to find the value of the expression to the nearest thousandth. (Calculator!!) ANSWER ≈≈ 1. log 5 21 ANSWER log 5 9 ANSWER log log ANSWER – Expand and Condense Logarithmic Expressions

Checkpoint Expand and Condense Logarithmic Expressions Expand the expression. Assume all variables are positive. 5. log 2 5x 6. log 2x 3 ANSWER log log x ANSWER log log 2 x 7. log 3 7 5x5x ANSWER log 3 5 – + log 3 xlog log 6 y 4x 24x 2 ANSWER log 6 4 – + 2 log 6 xlog 6 y

Checkpoint Condense the expression. Assume all variables are positive. ANSWER log 5 3 ANSWER log 2 35 ANSWER log log 5 12 log 5 4 – 10. log 2 7log log 42log log xlog y – ANSWER log y x 3x 3 Expand and Condense Logarithmic Expressions

1. Equal Powers Property 2. Equal Logarithms Property

Summary How do you solve logarithmic equations? If the equations can be rewritten so they have the same base and if a single variable appears as an exponent, take the logarithm of each side and solve for the variable; if the two sides of the equation can be written as logarithms to the same base, set the logarithms equal. Pg445 #(22-33, ODD) Pg452 #(27-33, ODD) Mid-Unit Quiz Monday/Tuesday UNIT Exam next week on Thursday/Friday. Assignment