math

graphing exponential func.

plug in usable terms
find asymptote: k-value
domain: all real numbers
range: k-value to (-)infinity
j curve

graphing logarithmic func.

inverse of exponential so swap coords
plug in usable terms
asymptote mirrors inverse k-value
domain: inverse k-value to (-)infinity
range: all real numbers

answer in the form of a, b, and c:
log((x^2*y^2)/z^3)

separate logs by taking exponent and putting it in front
then multiplication is addition and division is subtraction
substitute log(x) as a, log(y) as b, and log(z) as c

solve:
log2(x) + log2(x+1) = log2(6)

combine first two terms and drop logs
solve and factor quadratic from there

solve:
log9(81)+log9(1/9)+log9(3) = log(x)

combine first 3 terms and drop logs

solve:
log4(x+14)-log2(x)=1

change log4(x+14) to log2(x+14)/log2(4)
combine x's and evaluate log2(4) and combine with one
circle of logs and solve

solve:
log2(x) = log4(x+6)

change base of log4(x+6) to log2(x+6)/log2(4)
subtract log2(x+6) over and combine with log2(x)
solve log2(4) and circle of logs

solve:
ln(x)+1/ln(x)=2

substitute ln(x) as y
factor new equation
find y
set y equal to ln(x) and solve for x

solve:
2ln(x)^2 = 3ln(x)-1

substitute ln(x) as y
factor new equation
find y
set y equal to ln(x) and solve for x

solve:
3^(x+1) = 15

change to get like bases
drop bases
solve

solve and give answers in form of a + log(b):
2(10^(x+1)) = 15

divide by 2
log both sides
x+1log(10) cancels to x+1
subtract 1 to the other side
answer complete

solve and give answer in form of ln(a)/ln(b):
6^x = 3^(x+2)

ln both sides
separate x+2ln(3) to xln(3)+2ln(3)
subtract xln(3) to the other side
undistribute the x into the ln(6) and ln(3) to make x(ln(6)-ln(3))
divide by (ln(6)-ln(3))
get x = 2ln(3)/(ln(6)-ln(3))
simplify to ln(9)/ln(2)

solve and give answer in form of ln(a)/ln(b):
6^x-1 = 3^x+2

ln both sides
separate x-1ln(6) to xln(6)-1ln(6)
separate x+2ln(3) to xln(3)+2ln(3)
subtract xln(3) to other side
add 1ln(6) to other side
undistribute x to get x(ln(6)-ln(3))=1ln(6)+2ln(3)
divide by (ln(6)-ln(3))
get x = 1ln(6)+2ln(3)/(ln6)-ln(3))
simplify to ln(54)/ln(2)

solve:
4^x-9*2^x+8

subtitute 2^x as y
convert 4^x to 2^2x because math
write new equation: y^2-9y+8
factor and solve for y
set y equal to 2^x and solve for x

solve:
solve:
3^(2x+2)+8*3^(x+1)=9

substitute 3^x as y
get 9y^2+24y-9=0
factor and solve quadratic
set y = to 3^x and solve for x

graph e^(x-3)+2

find easy terms to put in and find ordered pairs:
(3,3) (4, e+2)
asymptote at y=2
domain: all real numbers
range 2 to infinity

find/graph inverse of e^(x-3)+2

swap x and y and solve for y
get y=ln(x-2)+3
swap ordered pairs from original
reflect over y=x
asymptote at x=2
domain: 2 to infinity
range: all real numbers

group of leopards growth is modeled by
N = 10e^(0.4t)

how many leopards after 2 years

how long until 100 leopards

how long until leopards double

first, plug in 2 for t and solve

second, set N = 100 and solve or calc it
for this divide by 10 then ln both sides
and divide by 0.4

third, set N = 20 and solve or calc it
for this divide by 10 then ln both sides and divide by 0.4

use FV = PV(1 + r/(100*n)^(nt)

invested 100

a: if interest compounded annually at 5%, write expression and find after 20 yrs

b: if interest compunted monthly at 5/12%, and this exceeds the answer to last problem in m months, write inequality and find m

first use formula: FV=PV(1+r/(100*n)^(nt)

a: first FV = 100(1.05)^t
second: plug in 20 for t

b: first set 256