Chapter 3 Quest
Base points of an exponential function
(0,1) and (1,b)
Base points for logarithmic functions
(1,0) and (b,1)
Compute the exact value of the function for the given x-value f(x) = -5*2^x for x = 1/3
-5*cbrt(2)
Exponential graphs always have
1 HA 2 points
Logarithmic graphs always have
1 VA 2 points
Find the exact solution algebriacally 2 * 2^x/3 = 32
1. Divide 2^x/3 = 16 2. Find out what 2 raised to = 16 2^4 = 16 2^x/3 = 2^4 x/3 = 4 x = 12 {12}
Tell whether the function is an exponential growth or decay and find the constant rate of crowth /decay f(x) = 4.8 * 1.05^x
1. Find r 1.05 = 1 +r r = .05 2.Compare r r>0 = true so function is expoential growth
State the domain of the function of f(x) = log[x(x+7)]
1. Find the Vertical asymptote x= 0 x = -7 2. Place into a sign chart + -7 - 0 + 3. Find the x intercepts: x(x+7) = 1 x^2 + 7x -1 = 0 quadratic formula x = 0.140 x = -7.140 4. State the domain (-inf , -7) U (0, inf) 5. Choose two random points to the left and right, and then plot them and draw curves
Graph the function below f(x) = -ln(x-5)
1. Identify Transformations Right 5 units, reflect across the x-axis 2. Identify VA x-5 = 0 x = 5 3. Apply to Base points of logarithms (1,0) --> (6,0) (e, 1) --> (e+5 , -1) 4. Graph
Express as a difference of logarithms log 66/y
1. Identify rule log(R/S) = log R - log S 2. log 66 - log y
Use the properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms log<5>^y^2
1. Identify the rule log<b> R^c = c * log<b> R 2. 2 * log<5> y
Use the properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms ln(25d)
1. Identify which propert log(R*S) = log R + log S 2. Split ln 25 + ln d 3. Reduce ln 5^2 + ln d 2 ln 5 + ln d
Find the exact solution algebriacally log<6>(2x-7) = 4
1. If solving a logarithm with a argument = to something 6^4 = 2x-7 1296 = 2x-7 2x = 1303 x = 651.5 {651.5}
1.23^x = 5.7
1. If solving and exponent is a variable log 1.23^x = log 5.7 x * log 1.23 = log 5.7 x = log 5.7 / log 1.23 x = 8.408 {8.408}
Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = e^x g(x) = e^-4x
1. List out transformations Horizonatly shrink by a factor of 1/4, Reflect over the y-axis 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (0, 1) (1,b) --> (1,e) --> (-1/4 , -e) 3.Graph
Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = 3^x g(x) = 3^x-3
1. List out transformations Right 3 units 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (3,1) (1,b) --> (1,3) --> (4,3) 3. Graph
Solve for x e^x - e^-x ------------ = 5 2
1. Multiply by 2 e^x - e^-x = 10 2. Multiply each term by [e]^x [e]^2x - 1 = 10[e]^x 3. Rearrange to a Quadratic [e]^2x - 10[e]^x - 1 4. Quadtratic formula [e]^x = Y Y^2 - 10Y - 1 Y= 5 +- sqrt(26) [e]^x = 5 + sqr(26) [e]^x = 5 - sqrt(26) ---> x can't be negative e^x = 5 + sqrt(26) ln 5 + sqrt(26) = x x = 2.312
State whether the function is exponential growth or decay and describe using limits f(x) = 1/3^-x
1. Must covert into a*b^x 1/3^-x 3^x 2. Compare to rules Growth: a>0 b>1 Decay: a>0 0<b<1 Exponential Growth lim x--> -inf f(x) = inf lim x--> inf f(x) = 0
Graph the function below f(x) = 4 + log(x)
1. Rewrite log(x) + 4 2. Identify transformations Up 4 units 3. Identify VA x = 0 3. Apply transformation to base points (1, 0) --> (1, 4) (10, 1) --> (10, 8) 4. Graph
Graph the function below f(x) = log(-3 - x)
1. Rewrite -(3 + x) --> -(x + 3) --> log(-x - 3) 2.Identify transformations Right 3, Reflect across y-axis 2.5 Identify VA -x-3 = 0 -x = 3 x = -3 3. Apply to base points of logartihms (1,0) and (b,1) (1,0) --> (-4,1) (10,1) --> (-13, 1) 4. Graph
Solve ln(3x-2) + ln(x-1) = 2 ln x
1. Simplify ln (3x-2 * x-1) = ln x^2 2. Same base logarithms = each other (3x-2 * x-1) = x^2 3x^2 - 5x + 2 = x^2 2x^2 - 5x + 2 = 0 (2x - 1) (x -2) x = 2, x = 1/2 (extraneous) {2}
Write the expression using only natural logarithms log<5> x
1. Use change of base ln x ---- ln 5
Write the expression using only common logs log<5> x
1. Use change of base log x ----- log 5
Describe how to transform the graph of ln x into the function given below f(x) = log<1/4> x
1. Use change of base to see how ln x turned into log<1/4> x ln (x) / ln (1/4) ln(x) / ln 1 - ln 1/4 --> ln(x) / -ln(4) -1/ln(4) * (ln x) 2. Identify Transformations Vertical shrink by a factor of 1/ln(4), and reflected across the x-axis 3. Identify VA: x = 0 4. Apply Transformations (1,0) --> (1 , 0) (e, 1) --> (e , -0.721) 5. Graph
Evaluate the logarithimic expression: log 1/10
1. Write as exponential 1/10 = 10^x 10^-1 = 1/10 log 1/10 = -1
Determine the formula for the exponential whose values are given (0,5/4) and (1,5/16)
1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 5/4 = a *b^0 (b^0 is 1) 5/4 = a 3. Solve for b using the other point and a plugged in 5/16 = 5/4 * b^1 1/4 = b^1 b = 1/4 4.Write as the function f(x) = 5/4 * 1/4^x
Determine the formula for the exponential whose values are given (0,7) and (2,14)
1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 7 = a *b^0 (b^0 is 1) a = 7 3. Solve for b using the other point and a plugged in 14 = 7 * b^2 2 = b^2 b = sqrt(2) 4.Write as the function f(x) = 7 * 2^x/2
Determine a formula for the exponential function with points (0, 2.5) and (1,4.7)
1. Write formula: P(t) = P(1 + r)^t 2. Plug in (0, 2.5) 2.5 = P(1+r)^0 P = 2.5 3. Plug in (1, 4.7) 4.7 = 2.5(1+r)^1 r = .88 4. Write as function P(t) = 2.5(1.88)^t
Evaluate the logarithimic expression: log<6> 1296
1. write as exponential 1296 = 6^x 6^4 = 1296 log<6> 1296 = 4
Compute the exact value of the function for the given x-value f(x) = 5*2^x for x = 0
5
In an exponential function there is 1 _____ but both ends ________ it
Horizontal Asymptote don't
Is the following function exponential? If so, state the initial value and base. If not explain why y = 5^z
Is an exponential function inital value is 1 (a) base is 5
Is the following function exponential? If so, state the initial value and base. If not explain why y = z^9
Not an exponential function because the base is a variable
Determine the exponential function that satisfies the given conditions Initial Value: 0.6 halving every 5 days
P(t) = .6(.5)^t/5 If being halfed every ___ # of days divide the t variable by it
Determine the exponential function that satisfies the given conditions Initial Value: 16 decreasing at a rate of 60% per year
P(t) = 16(1-.60)^t P(t) = 16(.4)^t
Determine the exponential function that satisfies the given conditions Initial Value: 5 increasing at a rate of 18% per year
P(t) = 5(1+.18)^t P(t)=5(1.18)^t
Exponential Modeling and conditions
P(t) = P(1+r)^t r>0 exponential growth r<0 exponential decay r must be a decimal
If log<b>U = log<b>V then
U = V
Evaluate the logarithimic expression: ln (e^-5)
Use ln e^n = n ln (e^-5) = -5
Change of Base -Uses and formula
Used to change logs with a base to common or natural logs log<b> N = (log<c> N) / (log<c> b)
Evaluate the logarithimic expression: log<11> 11
Using log<b> b = 1 log<11> 11 = 1
Solve log x^2 = 2
When dealing with solving, don't use exponential logarithm properties 10^2 = x^2 100 = x^2 x = +- 10
1/b^# is the same as
b^-#
Evaluate the logarithimic expression: 6^log<6> 2
b^log<b> m = m 6^log<6> 2 = 2
If a >0 and 0<b<1 the function is an ______________ function add limits
exponential decay lim x-->-inf f(x) = inf lim x--> inf f(x) = 0
If a > 0 and b > 1 the function is an ________________ function add limits
exponential growth lim x-->-inf f(x) = 0 lim x --> inf f(x) = inf
Exponetial function formula and conditions
f(x) = a * b^x a cannot = 0 b > 1
Evaluate the logarithimic expressionln ln 9throot(e^7)
ln (e^7)^1/9 ln (e^7/9) ln 9throot(e^7) = 7/9
Use the properties of logarithms to expand the logarithmic expression as much as possible log cbrt(x/z)
log (x/z) ^ 1/3 log (x^1/3 / z^1/3) log x^1/3 - log z^1/3 1/3 log x - 1/3 log z
Assuming x, y, z are positive use the properties of logarithms to write the expression as a single logarithm 8 log (xy) - 7 log(yz)
log(xy)^8 - log(yz)^7 log(x^8 y^8) - log(y^7 z^7) log(x^8 y^8) / (y^7 z^7) log(x^8 y) / z^7
Properties of all logarithmic functions
log<b> (R*S) = log<b>R + log<b>S log<b>(R/S) = log<b>R - log<b>S log<b> R^c = c * log<b> R
Properties of Log, Natural Logs, Common Logs
log<b> 1 = 0 ; ln 1 = 0; log 1 = 0 log<b> b = 1; ln e = 1; log 10 = 1 log<b> b^n = n; ln e^n = n; log 10^n = n b^log<b> m = m; e^ ln m = m; 10 ^ log m = m
logarithm arguments cannot be
negative
If b^u = b^v then
u = v
Solve for x log x = 2
x = 10^2 x = 100
when given an exponent of 1/x it is the same as
x root of base
Key to logartithms
y = a *b^x x = log<b> y
