Maths - Exponentials and Logarithms
What may the natural logarithm also be called?
The Naperian logarithm
What can the number in the logarithm (not the base) be called?
The argument
The larger the base of the exponential function...
The closer the curve comes to the y-axis
What does this mean in terms of graph transformations?
The graph of y = a^x reflected on the line of y=x = log(base a)x
What is the relationship between a^x and a^-x
The graphs are flipped in the y axis
What is meant by index form?
The opposite of logarithmic form, using exponents
What is a general rule for using bases in logarithms?
They cannot be negative
Since this is true, what can be said about the intersections of a logarithmic function its respective exponential function?
They will intersect on the line y=x
If there's no mathematical reason as to why a model may be unsuitable, what should be done?
Use a qualitative argument
Prove this formula
Suppose x = a^y log(base n)x = y log(base n)a y = log (base n)x/log(base n)a log(base a)x = log(base n)x/log(base n)a
How should you begin by solving some logarithms?
Take logs on both sides
What is the general equation for an exponential function?
y = a^x where a is a positive constant
How can this be shown?
y = a^x x = a^y log(base a)x = y y = log(base a) x
Give two popular relationships and their names
y = axⁿ y = ke^x
The most common base is...
10
What is the approximate value of e and what is the nature of the number
2.718 Irrational
What is the inverse of an exponential function?
A logarithmic function
Therefore, what is the derivative of the function e^kx
ke^kx
What relationship can therefore be stated?
ke^kx is directly proportional to y, where k is the constant of proportionality
Give four key points of logarithms
log (base 10) can be expressed as log a¹ = a therefore log (base a) (a) = 1 a⁰ = 1 therefore log (base a)(1)=0 a⁻¹ = 1/a therefore log (base a)(1/a) = -1
What is the second law of logarithms?
log(base a)(x/y) = log(base a)(x) - log(base a)(y)
What is the first law of logarithms?
log(base a)(xy) = log(base a)(x) + log(base a)(y)
What is the third law of logarithms?
log(base a)(xⁿ) = nlog(base a)(x)
Give the change of base formula for logarithms
log(base a)x = log(base n)x/log(base n)a
Construct a proof for the third law of logarithms
log(base a)xⁿ = log(base a)x1*x2*x3*x4...xn multiplication rule states that: log(base a)xⁿ = log(base a)x1 + log(base a)x2... Therefore = nlog(base a)x
What point do all exponential functions pass through and why?
(0,1) No matter the base, any number to the power of zero is one
What point do all logarithmic functions pass through and why?
(1,0) In any base, to get any number to equal one, you must raise it to 0
How else can a^-x be expressed
(1/a)^x
Why are exponentials often used as mathematical models?
Because the rate of change of some variable is proportional to the value of the same variable
When asked if data fits a mathematical model, what must be done?
Calculate percentage difference in data and mathematical value, base assumptions off of that
A base raised to an exponent and its logarithmic form can be said to be...
Equivalent statements
What are asymptotes of exponential and logarithmic functions?
Exponential: y≠0 (undefined) Logarithmic: x≠0 (undefined)
How do you solve logs when there are multiple operations?
Go from left to right
What is one reason a model may not appropriate?
If the variable is discrete
What happens to the shape of an exponential function as you increase the base?
It bends to the left
What happens to the shape of a logarithmic function as you increase the base?
It curves to the right
What is the name of ∞?
Lemniscate
How is this read?
O is the log of x to the base n
How would you model a compound interest after t years using an exponential function?
P(100+A/100)^t Where A is percentage and P is initial price
What can you do when you have logs on both the numerator and denominator?
Simplify them to common logs and cancel out
How do you prove that a model shows an upper limit
Simplify to get a whole number on the top, go from there
Give the standard logarithmic form of any base raised to an exponent
Where n°=x logₙ(x) = O
When a logarithmic problem is deduced to a quadratic/cubic, what must be kept in mind?
You cannot have a negative solution of x because you cannot take the negative of a log
Why can you not take the logarithm of a negative number?
You cannot raise a positive integer to something to get a negative number
These only work for what cases?
a > 0, a ≠ 1
Construct a proof for the second law of logarithms
a^n = x a^m = y log(base a)(x) = n log(base a)(y) = m x/y = a^n / a^m = a^(n-m) log(base a)(x/y) = n-m log(base a)(x/y) = log(base a)(x) - log(base a)y
Construct a proof for the first law of logarithms
a^n = x a^m = y log(base a)(x) = n log(base a)(y) = m xy = a^n * a^m = a^(m+n) log(base a)(xy) = m+n log(base a)(xy) = log(base a)(x) + log(base a)y
How can e be represented as a limit?
e = lim n-> ∞ (1+1/n)^n
Give a rule about the transformation of e^x
e^kx will compress the function by a scale factor of k However, for the same value of y, the transformed function will have a gradient of ke^x
What is the exponential function and why?
e^x The derivative of this graph is equal to itself
How else may e^x be written?
exp(x)
