Exponential and Logarithmic Functions
The Natural Base "e" (Euler's Number)
2.718281828459045235360287471 e is one of the most important numbers in mathematics. It is named after Leonard Euler. e is irrational and is the natural base of the logarithm. (1 + 1/n)ⁿ
exponential function
A function that is a transcendental function (not algebraic) because the transcend our ability to define them with a finite number of algebraic expressions. The variable is in the exponent and the constant is in the basis. An exponential function can be described by an equation of the form y= abⁿ, where b and x are any real numbers such that b > 0 and b ≠ 1 The domain is the set of all real numbers. example: f(x)=2ⁿ All go through the point (0, 1). All have x-axis as a horizontal asymptote.
compound interest
A=P(1 + r/n)^nt P=principal r=rate n=compounded times per year t=time in years
logarithmic function
An equation that involves a logarithm of a variable expression. An inverse of an exponential equation. Exponents and Logarithms work well together because they "undo" each other (so long as the base "b" is the same): y=logₙx is equivelant to x = n^y Example: log5 (4x-7) = log5 (x+5)
common logarithmic function
Another name for the logarithm with base 10. Often expressed as f(x) = log x. How many 10s do we multiply to get that number? Example: The common logarithm of 100 is 2(We need to multiply 2 10s to get 100) Example: The common logarithm of 1000 is 3
Carbon dating
Carbon dating was discovered by Libby in 1946 while working as a professor at University of Chicago. Carbon 14 is a radioactive variant of the significantly more common carbon 12. It is useful for dating objects that are 500 to 50,000 years old. It costs $600 to carbon date something. A = A₀e^(-0.000124t) t can be isolated by using the inverse.
Continuous Compound Interest
Compounds interest infinitely many times. This is the highest return you can earn. A = Pe^(rt) We can solve for t by getting the inverse: ln(A/P)/r=t. if A/P is a ratio e.g. P:3P then it simplifies to: t=ln(2)/r=0.7/r The "Rule of 70" = 70/100r
Decibels formula
D=10log(I/I₀) I₀ = The human average threshold = 1*10^-12
Real World Problems of Exponential Functions
Describe growth or decay. Archaelogists and anthropologists date fossils using carbon testing. The amount of carbon in a fossil is an exponential function of how many years the organism has been dead. Economics, investments e.g. compounding. Populations. Radioactive decay of isotopes. Used car depreciation.
Logarithmic functions are inverses of each other.
Doing one, then the other, gets you back to where you started. logₙn^x=x
Logarithmic functions are inverses of each other.
Doing one, then the other, gets you back to where you started. n^(logₙx)=x x>0
algebraic functions
Functions that involve basic operations, powers and roots. The variable is in the base and the constant is in the exponent.
natural exponential function
It is the exponential function f(x)=e^x with base e. It is often referred to as the exponential function.
Mental Math Logarithms
It is too bad they are written so differently ... it makes things look strange. So it may help to think of a^x as "up" and loga(x) as "down": Going up, then down, returns you back again:down(up(x)) = x Going down, then up, returns you back again:up(down(x)) = x
Power rule of logarithms
Log of a number raised to an exponent is the exponent times the log of the number. logₙx^y=ylogₙx
Product rule of logarithms
Log of a product is the sum of the logs.h logₙxy = logₙx + logₙy
Quotient rule of logarithms
Log of a quotient is the difference of the logs. logₙ(x/y) = logₙx - logₙy logₙ(1/y) = - logₙy
History of logarithms
Logarithms were very useful before calculators were invented. For example, instead of multiplying two large numbers, by using logarithms you could turn it into addition (much easier!) And there were books full of Logarithm tables to help.
Real World logarithmic functions
Measuring PH Value, decibels.
Exponential growth - World populations
N(t)=N₀e^rt N/N₀ = e^rt r = ln (N/N₀)/t t = ln(N/N₀)/r
irrational exponent
Numbers like this to be between rational exponents or "filling the gaps" from rational exponents.
Logarithms can only be evaluated for positive arguments.
True. log(-2) is undefined.
half life equation
Used for . A=A₀(1/2)^(t/h) A = Value at time t. A₀=Initial value h = half-life t = time when t=d, then P=2P₀
Doubling Time Growth Model
Used for when investments and population double at specific times. P=P₀2^(t/d) P=Population at time t P₀=Population at time t=0 d = Doubling time t = time when t=d, then P=2P₀
Change-of-base formula
Used to evaluate a logarithm if we cannot identify the exponent. logb(M)=loga(M) / loga(b) example. y=log₃8 log₃8 = log 8 / log 3 or ln 8 / ln 3 (either works)
natural logarithmic function
a logarithmic function with base e, often abbreviated as ln. It is how many times we need to use e in a multiplication to get our desired number. Examples: the natural logarithm of 7.389 is about 2, because 2.718282 ≈ 7.389 the natural logarithm of 20.09 is about 3, because 2.718283 ≈ 20.09
Evaluate exponential functions
f(x) = 2^x
Graph exponential functions
f(x) = b^x 1. Label the y-intercept (0,1) 2. Label two points f(-1) and f(1) note graph passes through (1,b) and (-1,1/b) 3. Connect the three points with a smooth curve considering that the x-axis is an asymptote. if b > 1 then the graph increases left to right. if b<1 (a rational number) then the graph decreases from left to right.
Transform graph of exponential functions - shifting
f(x)=2^(x-1) ; Shift graph right one. f(x)=2^x+1 ; Shift graph up one. Shift horizontal asymptote up one. f(x)=3*2^x ; vertically stretch. y-int increases to 3*1, so y int is now (0, 3). Horizontal asymptote does not change. f(x)=2^x/3; horizontal stretch, pulls out like a slinky. No change to y-int or horizontal asymptote.
Evaluate inverse exponential functions
f(x)=2^-x
evaluate natural exponent function
f(x)=e^x x=-0.47 f(x) = 0.6250
Find domain of a logarithmic function
f(x)=log(x²+3) https://www.youtube.com/watch?v=3GOpSXUJg2g
Natural Logarithm Properties
ln 1 = 0 ln e = 1 ln eⁿ = n e^(ln n) = n ln (0) is an error as n must be > 0 .
Common Logarithm Properties
log 1 = 0 log 10 = 1 log 10ⁿ = n 10^(log n) = n n >0
Definition of a logarithmic function and properties of exponentials.
logₙ1=0
Definition of a logarithmic function and properties of exponentials.
logₙn=1
PH Values in chemistry
pH = -log10([H+]) [H+] = 10-pH
graph natural exponent function
same as graphing exponent function
