Logarithms and Exponents

(a×)^y

a^xy

(ab)^×

a× • b^x → then multiply the two values found

5.2 Theorem on Graphs

The graph of f⁻¹(x) and f(x) are symmetric around the line y=x. Look for: • x-intercepts • y-intercepts • intesections • turning points

5.2 Theorem on Inverse Functions

(f⁻¹f(x)) = x and f(f⁻¹(x)) = x

A Base is a...

number raised to an exponent (ex. 5 in 5²)

a^(log↓aM)

M 5^(log₅2) = 5

a^(x/y)

y√a×

a× • a^y

a^x+y

a×/a^y

a^(x-y)

a⁻×

a^(1/x)

a⁰

1

APR

Annual Percentage Rate A=P(1+r)^t -rate as a decimal

Can you have negative logs or exponents?

You CANNOT take logs of negative numbers

Carrying Capacity

the maximum value that the function can attain

Change of Base Theorem

log↓aM = (logM/loga)

• Usually change log to ln

Common Log

logX = y if and only if x = 10^y

Compound Interest equation

A=P(1+r/n)^nt A= amount paid P= Principal r= rate as a decimal n= # of times compounded a year Year= 1 Quarterly= 4 Monthly= 12 Daily= 365

Continuously Compounded Interest equation

A=Pe^rt

Definition: The exponent base

e

Definition: y = log↓aX if and only if...

x=a^y

e = ....

2.7182818284

Effective Rate of Interest (section 5.7)

The equivalent annual simple rate of interest that would yield the same amount as compounding after 1 year

• Cane use Simple Interest or APR equation

Ex. If you want to earn $100 on a $1000 investment for 1 year

Ex. Change of Base Theorem: find log₅89

*2.789 (log89/log5) → (ln89/ln5)

Examples of Properties of Graphs when graphing (y=x² for this example)

• The y-intercept is always at (0

Exponential Functions

A function of the form y=a×

Exponential Line of Best Fit

y=ab^x

f is a One-to-One function if...

every horizontal line intersects the graph of a function at no more than one point

f(g(x))

Plug x in for g

f⁻¹(x) = x² is one-to-one if...

you restrict the domain to x≥0

Find the domain of y=log₂(1-x)

y=log₂(1-x) 1-x >0

• x|x<1

Finding Inverse Equations algebraically

Steps:

1. Replace f(x) with y.

2. Switch the letters x and y in the equation.

3. Solve the equation for x.

4. Rewrite the found equation

Formula for buying a home

P=L{(r/12)/(1-(1+r/12)^-t} P= monthly payment required to pay off the loan L= loan taken out r= annual rate of interest expressed as a decimal

Graph the function y=1/2× (flip of y=2×)

See section 5.3

Graph the function y=2×

See section 5.3

Graphs of logs: y=lnx

See section 5.4

Graphs of logs: y=log↓x

See section 5.4

Growth and Decay equation

A₀e^rt A= final amount A₀= initial amount K≠0 and is a constant t= time

Half-Life

The time for half of a radioactive substance to decay

If both are exponents on both sides...

take the log of both sides to solve

If f⁻¹ denotes the inverse of function f

then...

If the equation is f(x) = a×

where a>0

If trying to find a base or part of a log...

isolate the log and change to an exponent to solve

If you do not have the rate....

SOLVE FOR THE RATE FIRST!!!

In A=A₀e^rt

if k<0

In A=A₀e^rt

if k>0

In the logistic model formula

if b is negative

In the logistic model formula

if b is positive

In the logistic model formula

if b<0

In the logistic model formula

if b>0

Inverse functions of logs and exponents respectively are...

reflections over the line y=x

Is the following function exponential?

Domain: -1

Is the following function exponential?

Domain: -1

Is the following function exponential?

Domain: -1

Is this a one-to-one function? f(x) = x

Yes

Is this a one-to-one function? f(x) = x²

x⁴

Is this a one-to-one function? f(x) = x³

x⁵

ln =....

natural log

log↓a(1/n)

-log↓an

log↓a(M/N)

log↓aM - log↓aN

log↓a(MN)

log↓aM + log↓aN

log↓a1

x⁰ = 1

log↓aA

1 1¹=1 2²=1 3³=1 etc.

log↓aA^r

r log₁1⁵ = 5

log↓aM^r

rlog↓aM log↓x3⁵ = 5log↓x3

log↓aX is read as...

"log base a of x"

log₂8 is interpreted as...

2 to what power gives you 8?

Logarithmic Line of Best Fit

y=a+nlnx

Logarithmic One to One Property

log↓aM = log↓aN iff M=N b^M = b^N iff M=N

Logistic Line of Best Fit

y=c/1+ae^-bx

Logistic Model

Since there is usually a limit on how much something can grow

Logs are used to...

Solve for exponents

Natural Log

y = lnX if and only if x = e^y

Newton's Law of Cooling

U(t) = T+(U₀-T)e^kt U= Desired temperature U₀= Initial temperature T= constant temperature of the surroundings K= rate

One to One Function

A function is a one to one function if for every x there is exactly one y and one x for every y value.

Original vs. Inverse equations

f(x) vs. f⁻¹(x)

Present Value

The principal amount that you need to invest now so that it will grow to the given A value dollars in a specified time.

Ex. A bond can be redeemed in 10 years for $1000. How much will it cost you now if you can invest it for 8% compounded monthly or 7% compounded continuously?

Simple Interest equation

I = Prt I= interest P= Principal r= rate as a decimal t= time (usually in years)

Solve the following: 3׳ = 9×

3׳ = 9× 3׳ = (3)²×

• bases cancel x³ = 2x x³ - 2x = 0 x(x²-2) = 0 x = 0 x²-2 = 0 x² = 2 √x² = 2 |x| = √2

• x=0 x= ± 2*

Solve the following: 25× • 5ײ = 625²

25× • 5ײ = 625²

• change to equal bases 5²× • 5×₂ = 5⁸

• bases cancel 2x • x² = 8 x² + 2x - 8 = 0 (x+4)(x-2) = 0 x= -4 x= 2

Solve the following: e⁻ײ = e²× • 1/e³

e⁻ײ = e²× • 1/e³ e⁻ײ = e²× • e⁻³ -x² = 2x -3 -x² - 2x + 3 = 0 (x+3)(x-1) = 0 x=-3 x=1

Solve the following... 3^(x+1) = 81 using change of base!

3^(x+1) = 81 3^(x+1) = 3⁴

• bases (threes) cancel x+1 = 4 x = 3

Solve without a calculator using change of base: log₂4 • log₄6 • log₆8

*3 (log4/log2) • (log6/log4) • (log8/log6)

• cancel out log8/log2

• change to log function log₂8 = 3

Solve: 10× = 14.5

10× = 14.5 log₁₀14.5 *1.161

Solve: e^b = 9

e^b = 9 ln(9) = b *b= 2.197

Solve: log₁.₂M = 3

log₁.₂M = 3 1.2³ = M *M= 1.728

Solving using U: solve 4^x - 2^x - 12 = 0

4^x - 2^x - 12 = 0

• let u be 2^x 4^x = (2²)^x = (2^x)² = u² 2^x = u u²-u-12 =0

• Solve like a regular quadratic (u-4)(u+3) =0 u=4 u=-3

• Substitute u back in to solve for x 2^x=4 2^x=-3 x=2 can't have negative *x=2

Strategies for solving exponential and logarithmic equations: If trying to find an exponent...

get the exponent by itself and change to a log to solve

Strategies for solving exponential and logarithmic equations: Use the...

• Change of base formula or other properties to solve (see section 5.5)

• One to One Property

The domain of the function f(x) = a×

where a>0

The graph of e is...

See section 5.3

The range of the exponential function f(x) = a×

where a>0

Theorem: For an exponential function f(x)=a×....

(f(x+1)/(f(x))=a

if the x values increase by equal intervals and the y values increase exponentially equally

then it is exponential.

To determine the amount of interest paid from a final amount...

Subtract the initial amount paid from the final amount being paid (see MathLab 5.7

To find the domain of a log with (x-1) for example

do this...

Using Transformations to graph exponential functions: y=2⁻× - 8

where does it limit?

Reflect across the x-axis and translate down 8 units

limits at y=-8

When finding lines of best fit in your calcuator or finding k(rate) values...

KEEP THEM CARRIED ALL THE WAY OUT!

• Store line equations into y=

• Store K values as A

Write 5² = 25 as a log

2 = log₅25