Algebra 2 Honors - Unit 6 (Exponents and Logarithms)


6.1a - Key Features of Exponential Functions

Exponential: the function f(x)=bxf\left(x\right)=b^{x} , b > 0, b is not equal to 1, and b and x are real numbers, is called an exponential function base b

  • Exponential “growth” is y=2xy=2^{x}

  • Exponential “decay” is y=(12)xy=\left(\frac12\right)^{x}

  • They’re reflections of each other over the x-axis

  • When graphing, find the points that correspond with x = -1, 0, 1

    • Ex: For f(x)=5xf\left(x\right)=5^{x} , the points we need are (-1, 1/5), (0,1), and (1, 5)


Exponential growth: A(t)=a(1+r)tA\left(t\right)=a\left(1+r\right)^{t} where a > 0, b > 1, and b = 1 + r

  • 1 + r = b, and b must be > 1 to indicate growth

Exponential decay: A(t)=a(1−r)tA\left(t\right)=a\left(1-r\right)^{t} where a > 0, 0 < b < 1, b = 1 - r

  • 1 - r = b, and b must be < 1 to indicate decay

IMPORTANT:

  • a = initial amount

  • r = rate

  • t = number of time periods

Exponential function: y=A⋅Bxy=A\cdot B^{x}


PRACTICE WRITING EXPONENTIAL MODELS LIKE IN THE NOTES WORKSHEET


6.1b - Key Features of Exponential Functions

PRACTICE QUESTIONS YOU CAN TRY


6.2 - Exponential Models/Natural Base


Compound Interest: the amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year

Formula: A=P⋅(1+rn)ntA=P\cdot\left(1+\frac{r}{n}\right)^{nt}

Euhler’s Number/natural base (e): an irrational number that the expression (1+1n)n\left(1+\frac{1}{n}\right)^{n} approaches as n→∞n\to\infty

Continuous Growth Model/Law of uninhibited growth/Law of exponential growth: Used for interest problems when the interest is compounded continuously

Formula: A=PertA=Pe^{rt}

  • p = present value

  • r = growth rate

  • t = annual time period


PRACTICE THE QUESTIONS IN THE NOTES AND STUFF


6.3/6.4 - Logarithmic Functions

THIS IS ALL ABOUT GRAPHING LOGARITHMIC FUNCTIONS AND INVERSES TOO SO JUST REFERENCE NOTES


6.3-6.4b - Logarithmic Functions

For all b > 0, and b is not equal to 1:

y=bx⇔x=log⁡byy=b^{x}\lrArr x=\log_{b}y

Theorems:

  1. If au=ava^{u}=a^{v} then u = v (just make the base the same, and you don’t need to check for extraneous solutions)

  2. y = log x if and only if x = 10y

  3. y = ln x if an only if x = ey

  4. log⁡bbx=x\log_{b}b^{x}=x and blog⁡bx=xb^{\log_{b}x}=x

    1. Ex: 4log⁡43=34^{\log_43}=3 and log⁡22π=π\log_22^{\pi}=\pi


6.3-6.4c - Logarithmic Functions

  1. log⁡b(X)+log⁡b(Y)=log⁡b(XY)\log_{b}\left(X\right)+\log_{b}\left(Y\right)=\log_{b}\left(XY\right)

  2. log⁡b(X)−log⁡b(Y)=log⁡b(XY)\log_{b}\left(X\right)-\log_{b}\left(Y\right)=\log_{b}\left(\frac{X}{Y}\right)

  3. log⁡b(AP)=Plog⁡b(A)\log_{b}\left(A_{}^{P}\right)=P\log_{b}\left(A\right)


How to Estimate a log with log 750:

  1. Determine two powers of 10 that 750 is between

    1. 100 < 750 < 1000

  2. Determine the two integers that log 750 is between

    1. log 100 = 2 < log 750 < log 1000 = 3

  3. Therefore a good estimate of log 750 is 2.87


6.5 - Properties of Logarithms

Formulas:

  1. log⁡bx=log⁡axlog⁡ab\log_{b}x=\frac{\log_{a}x}{\log_{a}b}

  2. Half life: y=a(12)tn=Pe(ln⁡12n)ty=a\left(\frac12\right)^{\frac{t}{n}}=Pe\left(\frac{\ln\frac12}{n}\right)^{t}

  3. Double: yy=a(2)tdy=a\left(2\right)^{\frac{t}{d}}


PRACTICE PROBLEMS YOU CAN TRY ON THE NOTES WORKSHEET


6.6a - Exponential Equations

PRACTICE PROBLEMS YOU SHOULD TRY ON THE NOTES WORKSHEET


6.6b - Solving Logarithmic Equations

PRACTICE PROBLEMS YOU SHOULD TRY ON THE NOTES WORKSHEET


Topic 6 Quiz #1 and 6.6c - Applications Exponential and Logarithmic Equations

PRACTICE PROBLEMS YOU SHOUlD TRY ON THE NOTES WORKSHEET