7.6 Objective: Students will solve exponential and logarithmic equations.

7.6Objective: Students will solve exponential and logarithmic equations. Warm-up: page 505 (64-79)

Review of Rational Exponentsand Introduction to Exponential Functions

The exponent laws can be extended to help us understand rational exponents. and Same Therefore:

Expressing Powers as Radicals The base becomes the radicand. The numerator becomes the exponent. The denominator becomes the index.

Using Radicals to Evaluate Powers = 8 Exponential Notation Radical Form Standard Form = 4 = 9 = (-3)2 = (5)3 = 125 = (4)5 = 1024

Evaluating Powers with Negative Exponents and Bases = imaginary By definition, we say that: Therefore: = -64i

Expressing Radicals as Powers

y = 4x y y = 3x y = 2x x Exponential Functions What about y=(1/2)x ? Graph the curves y = 2x y, = 3x, y = 4x on the same coordinate system y=(1/2)x 1

Properties of the basicy = bx function • The domain consist of all real numbers x • The range consists of all positive all real numbers y • The function is increasing when b>1 and decreasing when 0 <b < 1 • It is one to one function • The x –axis is the horizontal asymptote to curve, toward the left when b>0 and toward the right for 0<b<1

Exponential Function: Any equation in the formf(x) = Cbx. if 0 < b < 1 , the graph represents exponential decay – the y-value is going down as x increases if b > 1, the graph represents exponential growth – the y-value is going up as x increases Examples: f(x) = (1/2)x f(x) = 2x Exponential Decay Exponential Growth We will take a look at how these graphs “shift” according to changes in their equation...

Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x : Vertical Shift: The graphs of f(x) = Cbx + k are shifted vertically by k units. f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3

Take a look at how the following graphs compare to the original graph of f(x) = (2)x : (3,1) (0,1) (-2,-2) Notice that f(0) = 1 Notice that this graph is shifted 3 units to the right. Notice that this graph is shifted 2 units to the left and 3 units down. f(x) = (2)x f(x) = (2)x – 3 f(x) = (2)x + 2 – 3 Horizontal Shift: The graphs of f(x) = Cbx – h are shifted horizontally by h units.

Take a look at how the following graphs compare to the original graph of f(x) = (2)x : (0,1) (0,-1) (-2,-4) Notice that f(0) = 1 This graph is a reflection of f(x) = (2)x . The graph is reflected over the x-axis. Shift the graph of f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3

y y = 1+2x 2 y = 2x y = 2x-3 x X =3 Example Use the graph of y = 2x to sketch the curves y = 2x-3 y = 2x+1 1

Exponential Equations • One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. • For b>0 & b≠1 if bx = by, then x=y

Solve by equating exponents • 43x = 8x+1 • (22)3x = (23)x+1 rewrite w/ same base • 26x = 23x+3 • 6x = 3x+3 • x = 1 Check → 43*1 = 81+1 64 = 64

Your turn! • 24x = 32x-1 • 24x = (25)x-1 • 4x = 5x-5 • 5 = x Be sure to check your answer!!! Try: page 515 (1-3)

Assignment Page 519 (3-11 all) Page 485 (mixed review) 45-64

7.6 Objective: Students will solve exponential and logarithmic equations.