Warm Up Evaluate the following. 1. f(x) = 2 x when x = f(x) = log x when x = f(x) = 3.78 x when x = f(x) = ln x when x = f(x) = -5 x when x = f(x) = log 4 x when x = 1024

Graphing Exponentials and Logarithms March 18, 2013

Objective SWBAT identify the graphs of exponential and logarithmic functions SWBAT graph exponential and logarithmic functions through transformations

Vocabulary Exponential Growth Exponential Decay Domain of a Log An exponential function that is always increasing An exponential function that is always decreasing Set the inside of the log ≥ 0 and solve for x

Exponential Growth Basic Exponential Function f(x) = a x, a > 1 Domain: All Real Numbers Range: Positive Numbers Intercept: (0, 1) Increasing Horizontal Asymptote: y= 0

Exponential Decay Basic Exponential Function f(x) = a -x, a >1 Domain: All Real Numbers Range: Positive Numbers Intercept: (0, 1) Decreasing Horizontal Asymptote: y= 0

Logarithmic Basic Logarithmic Function f(x) = log a x, a > 1 Domain: Positive Numbers Range: All Real Numbers Intercept: (1, 0) Increasing Vertical Asymptote: x = 0 Reflection of y = a x across the line y = x

Identifying Type of Function Look for intercepts and asymptotes Look for increasing or decreasing behavior

Example 1.2.

Practice

Solving Exponential Equations

Objective SWBAT solve exponential and logarithmic equations

Vocabulary Exponential Equation Logarithmic Equation y = b x x = log b y

Solving Exponential Equations 1.Simplify the equation so the exponential is isolated on one side of the equal sign 2.Rewrite the equation as a logarithm using the definition 3.Decide if an exact answer or an approximate solution is preferable

Solving Simple Equations In general there are two strategies for solving exponential and logarithmic equations, look for opportunities to use the one-to-one property or use the inverse property. In either case, first rewrite the equation to see which property should be used then solve for x

Example 1.2 x = 32

Example (1/3) x = 9

Example 1.3(2 x ) = 42

Example 2. e x + 5 = 60

Example 3. 2(3 2t-5 ) – 4 = 11

Practice 1.e x = x = x = 1/9 4.2(5 x ) = e x = x-3 = 32 7.e 2x = e x = (3 6-2x ) + 13 = – 3(4 2x-1 ) = (9 3x+8 ) = (e 7x+1 ) – 9 = 7

Solving Logarithmic Equations Simplify the equation so the logarithm is isolated on one side of the equal sign Rewrite the equation as a exponential using the definition Decide if an exact answer or an approximate solution is preferable

Example 1. ln x = -3

Example log 10 x = 2

Example log 4 (3x) = 4

Example 3 log 5 (x + 1)= -6

Practice 1.ln x = 7 2.log 4 x = log 5 x = ln x = 16 5.log 2 (x – 3) = 3 6.ln (2x) = ln x = log 3 (6-2x)+13=35 9.6–3log 4 (2x–1)= ln(7x + 1) – 9 = 7