Warm-up

Logarithmic Functions 8.4 Logarithmic Functions

You know that 22 = ________ and 23 = _______ You know that 22 = ________ and 23 = _______. But do you know the value of x that satisfies the equation 2x = 6? We can guess that the value of x is between ________ and ________, but to find the exact value, we use something called a logarithm.

Definition of Logarithm with Base b If bx = y, then logb y = x. x = exponent b = base Read as: log base b of y

Example 1: Write the equation in EXPONENTIAL FORM. a) log3 9 = 2 b) log8 1 = 0 c) log5 1/25 = -2 d) log1/2 2 = -1

Example 2: Evaluate each expression. a) log4 64 b) log2 1/8 c) log 1/4256 d) log32 2

Common Logarithm: log with base 10 Notation: log x If no base is written, assume the base is 10. Natural Logarithm: log with base e Notation: ln x

Example 3: Use your calculator to evaluate. a) log 7 b) ln 0.25 c) ln 0.1 d) log 10

Logarithm of 1 Logarithm of base b Special Logarithm Values- Let b be a positive real number such that b is not 1. Logarithm of 1 Logb 1 = 0 because b 0 = 1 (Doesn’t matter what your base is) Logarithm of base b Logb b = 1 because b 1 = b Same numbers

Finding Inverses Example 5: Find the inverse of each function. a) y = log4 x b) y = ln (x + 1) c) y = log (x - 3)

Teaching Example Find the inverse of y = logb x Notice: Exponential and Logarithmic functions are inverses!

Let f(x) = bx and let g(x) = logb x. Using Inverses to Simplify Expressions Let f(x) = bx and let g(x) = logb x. Verify that these functions are inverses.

Properties! Our verification helped us find properties. 1.) 2.)

Example 6: Simplify each expression. a) 4log4x b) 10logx c) 9log9x d) log327x

Warm-up for Day 2 Rewrite in exponential form. 1. Evaluate the expression without a calculator. 2. 3. Use a calculator to evaluate. Round to 3 decimal places. 4. ln10 5. log0.3 Simplify 6. 7.

Warm-up for Day 3 Rewrite in exponential form. 1. Evaluate the expression without a calculator. 2. 3. Use a calculator to evaluate. Round to 3 decimal places. 4. log3.5 5. ln22.5 Simplify 6. 7.

Graphing logarithmic functions The graph of y = logb (x – h) + k has the following characteristics: x = h is a vertical asymptote The domain is x > h, the range is all real numbers If b > 1, the graph moves up and right If 0 < b < 1, the graph moves down and right

Steps to graph Determine the asymptote and sketch it in. Find convenient points using the un-shifted graph. Make an xy-table Hint: Always use x = 1 as one of the points. Make a new xy-table applying the correct shifts. Left/right changes the x-values Up/down changes the y-values Plot new points Determine domain and range of graph.

Example 7: Graph each function. State the domain and range. a) y = log5 (x + 1) x

Example 8: Graph each function. State the domain and range. b) y = log1/4 (x – 2) – 3 x