Logarithms and exponential functions Flashcards
Properties of Exponents - multiplication
a ^m x a^n = a ^m+n
Properties of Exponents - division
a^m / a^n = a^m-n or 1/a^n-m
Properties of Exponents - a ^0
a^0 = 1
Properties of Exponents - (a^n)^m
a ^n x m
Properties of Exponents - (ab)^m
a ^m x b^m
Properties of Exponents - (a/b)^n
a^n/b^n
Negative Exponents
a ^-n = 1/a^n
fraction over fraction
1/1/2 = 2/1
Do reciprocal of denominator
Properties of Exponents - a ^ 1/2
a ^ 1/2 = square root of a
Properties of Exponents - a ^ 1/3
a ^ 1/3 = cubed root of a
Properties of Exponents - a ^ m/n
n (square root of) a^m
how to solve for x (2 ways)
- make the base the same, then equate the exponents
- change to logarithmic form and solve (either with calculator or by hand)
general exponent function
y = a^x
logarithmic form
log(base) (result) = Exponent
relationship between logs and exponentials
inverse functions
Properties of logs - multiplication
- Log (a x b) = log (a) + log (b)
Properties of logs - division
log (a/b) = log (a) - log (b)
Properties of logs - exponent
log (a ^ n) = (n)log (a)
if log has no base
it means its a base 10
how to solve logarithmic functions
- turn into an exponential equation
- exponentiate both sides with a base. (if its log 2, make it 2 ^log 2) so that log cancels out with base.
Log e (x)
natural log, written as ln (x)
eulers constant
how to find inverse
switch x and y, turn to log, then change back
how to find initial area
make exponent 0
how to find at an x amount of weeks
make exponent x
how to show growth
positive exponent
how to show decay
negative exponent
example: the amount of rubbish in the tip is A = 3000 x 1.05^n because it increases by 5% (0.05). Why is it 1.05
because it increases by 5% of the initial value, which can be written as 3000 + (3000 x 0.05). This can be rewritten as 3000 + (1+ 0.05) or
3000 + 1.05
