Section+3.3

=3.3 Logarithmic Functions and their Graphs=  Example: 8 = 2^x 2^3 = 2^x 3 = x We are able to break down 8 into 2^3 in order to give us a common base on each side.  Example: 5 = 2^x ? = xWe are unable to break down 5 to give us a common denominator, which is why there is a ?. Find the inverse of y = 2^x x = 2^y Solve for y? __** Definition :**__ If b, x>0, b does not equal 0, then y=log(b)x if and only if b^y=x. Note: y=log(b)x is logarithmic form, b^y=x is exponential form.  Example: Evaluate log(2)8 log(2)8 = x 2^x = 8 2^x = 2^3 x = 3

 Example: log(3)sqrt3 log(3)sqrt3 = x 3^x = sqrt3 3^x = 3^1/2 x = 1/2

Example: log(5)1/25 log(5)1/25 = x 5^x = 1/25 5^x = 25^-1 5^x = (5^2)^-1 5^x = 5^-2 x = -2

Example: log(4)1 log(4)1 = x 4^x = 1 x = 0

Example: log(7)7 log(7)7 = x 7^x = 7 x = 1

__**Properties: **__ For x,b>0, b does not equal 0, y is an element of R: 1. log(b)1 = 0 2. log(b)b = 1 3. log(b)b^y → log(3)9 = log(3)3^2 = 2 4. b^log(b)x = x → 6^log(6)11 = 11

__**Common Base: **__ Base 10 (usually not shown) Example: log10 = 1 Example: log1000 = 3 Example: log1/100 = -2 Example: Solve logx = 3 10^3 = x 1000 = x Example: Solve log(2)x = 5 2^5 = x 32 = x Base "e" log(e)=ln Note: log(e) is BAD, ln is GOOD! Example: ln sqrt(e) = 1/2 <span style="color: rgb(246, 19, 19);">Example: ln(e^5) = 5<span style="color: rgb(248, 13, 13);"> <span style="color: rgb(245, 15, 15);">Example: e^ln^4 = 4 <span style="color: rgb(249, 6, 6);">Example: ln23.5 = 3.157
 * <span class="Apple-style-span" style="color: rgb(8, 12, 212);">__Natural Base:__ **

** ** Inverse functions of exponential functions, so they have symmetry over y=x line. ** <span style="color: rgb(246, 14, 14);">Example: Graph- y = log(2)x Log Growth Inverse: 2^x = y The graph created is a curve going through the point (1,0) with the "y" values approaching negative infinity and the "x" values approaching positive infinity. <span style="color: rgb(249, 11, 11);"> <span style="color: rgb(241, 9, 9);">Example: Graph- y = log(1/2)x Log Decay Inverse: (1/2)^x = y The graph created is a curve going through the point (1,0) with the "y" values approaching positive infinity and the "x" values approaching positive infinity. __**<span style="color: rgb(4, 19, 220);">Family of Functions :**__ f(x) = a•log(b)(x-h)+k **Properties:** 1. b>1 Growth 0<b<1 Decay 2. x=0 is a vertical asymptote 3. x-intercept is (1,0) 4. Domain: (0, positive infinity) Range: all real numbers 5. One-to-One <span style="color: rgb(245, 5, 5);">Example: f(x)=-2log(2)(1-x)+4 From this equation, we can see that the graph will be transformed up 4 units, right one unit, vertically stretched by 2, and reflected over the x and y axis. We can also tell this it is logarithmic by the log sign, and the base 2 tells us that it is growth. <span class="Apple-style-span" style="white-space: pre; color: rgb(19, 235, 10);">**HOMEWORK PROBLEMS:** pg. 308, #3-60 (mult. of 3) pg. 308, #1-58 (by 3's)
 * <span class="Apple-style-span" style="color: rgb(7, 17, 213);">__Graphing:__