Exponential and Logarithmic Functions

Graphing

100

What would be the transformation of 3^x+1

It translated one unit up

200

What is the x-intercept of all non translated exponential graphs

(0,1)

300

Use the graph of f to describe the transformation that yields the graph of the function. f(x)= (7/2)^x, g(x)= -(7/2)^-x+6

Reflect the graph of f in the x-axis and y-axis and shift six units to the right.

400

True or False, the line y= -2 is an asymptote for the graph of f(x) = 10^x - 2. Justify your answer.

true because it will go left until for negative infinity, but it will get so close to -2 but never touch, making -2 an asymptote.

500

Find the Domain, x-intercept, and vertical asymptote of the logarithmic function and describe the graph. g(x) = ln(-x)

Domain: (-infinity, 0) x-intercept: (-1,0) Vertical asymptote: x=0 The Graph will come the upper left quadrant and pass through (-1,0) and then travel close the x=0 till infinity without touching it.

Evaluating the natural Exponential Function

100

Evaluate the function given by f(x) = e^x when x=-2

f(-2) = e^-2 = 0.13533553

200

h(x) = e^-x    x = 3/4

0.472

300

f(x) = 5000e^0.06x  x=6   

7166.647

400

A total of 12000 is invested at an annual interest rate 9%. Find the balance after 5 years if it is compounded. Use the formula A=Pe^rt

=12,000e^0.09(5) =18,819.75

500

With the given formula for compounding per year: A=P(1+r/n)^nt. A total of 12000 is invested at an annual interest rate 9%. Find the balance after 5 years if it is compounded. Find the quarterly compounding by setting n=4.

A=P(1+r/n)^nt = 12000(1+ 0.09/4)⁴⁽⁵⁾ approximately equal to 18,726.11

Logarithmic Functions

100

Use the definition of logarithmic function to evaluate each logarithmic at the indicated value of x. f(x) = log ₂x    if x = 32

f(32) = log₂32 = 5 because 2⁵ = 32 

5

200

Use the definition of logarithmic function to evaluate each logarithmic at the indicated value of x. f(x) = logx, x=1/100.

f(1/100)= log1/100=-2 -2

300

evaluate for the function f(x) = logx     x=-2

No solution, no real number power to which 10 can be raised to obtain -2.

400

Use the one to one property to solve log(x² - 6) = log10.

x² -6 = 10     x² = 16      x= ⁺/_-  4

500

ln((x²) -2  = ln 23 

Use the One to One property to solve for x.

x=-5,5

Properties of Logarithms

100

Use the changing base formula to solve log₄ 25

log 25/log 4 

2.3219

200

use the natural logarithms changing base formula to solve log ₂ 12.

ln 12/ln 2 =3.5850

300

Use the properties of logarithms to write the logarithms in terms of ln2 and ln3 : ln2/27

=ln2 - ln27 =ln2 - ln3^3 =ln2-3ln3

400

Condense the expression to the logarithm of a single quantity. 1/4log₃ 5x

log₃ 5x to the fourth root

500

logx - 2logy + 3logz. condense to a single quantity.

 logxz³/y²

Exponential and Logarithmic Equations

100

Solve. e^x=7

lne^x = ln7      x=ln7

200

e^x +5 = 60 Solve.

e^x=55

lne^x  =  ln55     x=ln55  =  4.007

300

2(3^2t-5) - 4 = 11 Solve.

t = 3.417

2(3^2t-5)=15

3^2t-5= 15/2      log₃3^2t-5=log₃ 15/2  

2t-5 =log₃7.5   t =5/2 + 1/2 log₃ 7.5 

400

Solve. 5 + 2lnx =4

x=0.607

5+2lnx=4      2lnx=-1    lnx=-1/2    e^lnx=e^-1/2 x=e^-1/2

500

You have deposited $500 in account that pays 6.75% interest, compounded continuously. how long will it take to double? Use the formula A=Pe^rt

t=10.27

A=500e=1000  e^0.0675t= 2 lne^0.0675t=ln2 0.0675t=ln2 t=ln2/0.0675