What is a log?
It is the inverse of an exponent
What question does a log answer?
“To what power must a base be raised to get a certain number”
loga(XY)
logaX + logaY
What is the product rule of logarithms?
loga(XY) = logaX + logaY
The log of a product is the sum of the logs of its elements.
loga(X/Y)
logaX - logaY
The log of a quotient equals the difference between the logs of the numerator and the denominator
What is the quotient rule of logs?
loga(X/Y) = logaX - logaY
The log of a quotient equals the difference between the logs of the numerator and the denominator
loga(1)
= 0
What is the zero rule of logs?
loga(1) = 0
The log of 1 to any base is always 0.
loga(a)
= 1
What is the identity rule of log?
The log of a base to itself is always 1.
Formula: loga(a) = 1
Example: log7(7) = 1
loga(1/X)
= -loga(X)
The log of a number’s reciprocal (1/number) is equal to the negative of the log of the number.
What is the reciprocal rule of logs?
The log of a number’s reciprocal (1/number) is equal to the negative of the log of the number.
Formula: loga(1/X) = - loga(X)
Example: loga(1/2) = - loga(2)
loga(X^n)
n x loga(X)
The log of a number raised to an exponent equals the exponent multiplied by the log of the base.
Formula: loga(X^n) = n × logaX
Example: log5(9^2) = 2 × log5(9)
What is the power rule or exponential rule of log?
The log of a number raised to an exponent equals the exponent multiplied by the log of the base.
Formula: loga(X^n) = n × logaX
Example: log5(9^2) = 2 × log5(9)
Apply the change of base or the base switch rule to loga(X)
loga(X) = logᵦ(X) / logᵦ(a)
What is the change of base of log rule or the base switch rule?
Formula: loga(X) = logᵦ(X) / logᵦ(a)
Example: log3(7) = log10(7) / log10(3)
This rule enables one to calculate the log of a number in a different base (typically done for base 10 or base e)
loga(a^n)
Formula: loga(aⁿ) = n
Example: log₄(4²) = 2
What is the log inverse property?
Calculating the log of an exponentiated value yields the original exponent.
Formula: loga(aⁿ) = n
Example: log₄(4²) = 2
x^1
x^1 = x
6^1 = 6
x^0
x^0 = 1
8^0 = 1
x^-1
x^-1 = 1/x
6^-1 = 1/6
(x^m)(x^n)
(x^m)(x^n) = x^(m+n)
(4^1)(4^3) = 4^(4)
x^m / x^n
x^m / x^n = x ^(m-n)
4^7 / 4^1 = 4^6
(x^m)^n
(x^m)^n = x^mn
(x^2)^3 = x^6
