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Logarithms and Log Properties Algebra Cheat Sheet
Definition Logarithm Properties
y logb==1log10y==logx is equivalent to xb bbbBasic Properties & Facts
x logxb logbxArithmetic Operations Properties of Inequalities b
Example If a0 then ac<
loglocommon log 10 cŁł Properties of Absolute Value where e=2.718281828K The domain of logx is >0 b
acad+-bcaadbc aaif 0‡
+=-= Factoring and Solving a=
bdbdbdbd -
22c--ddcccc a 22a -b–-b4acx+2ax+a=+xa x= b==ab a bba2x-ax+a=-xa 2()ab+ ac b adŁł If b->40ac - Two real unequl solns. a+b£+abTriangle Inequality=b+ca,0„= 2 2ca bc x+(a+b)x+ab=x++a)(xb ) If b-=ac - Repeated real solution. 23d 3223 If b-<40ac - Two complex solutions. Łł Distance Formula x++33axax+a=+(xa)
Exponent Properties If P=xy, and P=xy, are two ()()111 222 32n x-ax+ax-a=-xa Square Root Property a 1nmn+-mnm=aa== points the distance between them is 2mmn- 3322 If xp= then xp=– x+=ax+ax-+axa() nnm 0 22aa=„1, 0 3( ) dP,Px-x+-yy x-a=x-ax++axa( ) ( ) ( ))122121 Absolute Value Equations/Inequalities
n n 22nnnnnn If b is a positive number nnn x-a=x-+axa ()((ab)==abn p=b=p-=bor pbComplex Numbers bbŁł If n is ...
Logarithms and Log Properties Algebra Cheat Sheet
Definition Logarithm Properties
y logb==1log10y==logx is equivalent to xb bbbBasic Properties & Facts
x logxb logbxArithmetic Operations Properties of Inequalities b
Example If a0 then ac<>bc and log=-logloglnxx=lognatural log== becc y Łłcbcb b
loglocommon log 10 cŁł Properties of Absolute Value where e=2.718281828K The domain of logx is >0 b
acad+-bcaadbc aaif 0‡
+=-= Factoring and Solving a=
bdbdbdbd -40ac - Two real unequl solns. a+b£+abTriangle Inequality=b+ca,0„= 2 2ca bc x+(a+b)x+ab=x++a)(xb ) If b-=ac - Repeated real solution. 23d 3223 If b-<40ac - Two complex solutions. Łł Distance Formula x++33axax+a=+(xa)
Exponent Properties If P=xy, and P=xy, are two ()()111 222 32n x-ax+ax-a=-xa Square Root Property a 1nmn+-mnm=aa== points the distance between them is 2mmn- 3322 If xp= then xp=– x+=ax+ax-+axa() nnm 0 22aa=„1, 0 3( ) dP,Px-x+-yy x-a=x-ax++axa( ) ( ) ( ))122121 Absolute Value Equations/Inequalities
n n 22nnnnnn If b is a positive number nnn x-a=x-+axa ()((ab)==abn p=b=p-=bor pbComplex Numbers bbŁł If n is odd then,
nnn-1nn--21p< -<bbor pbxa+ a+bi+c+d=a+c++bdin ( ) ( ) ( )n 1abbn1 mnmm n-1n-2231==a==aa( )( ) =x+axax+a-+L () )a+bi-c+di=a-c+-bdi( ) ( ) ( )baaŁłŁł
Completing the Square a+bic+di=ac-bd++adbci( )( ) ( )
2 (4) Factor the left side Properties of Radicals Solve 2xx-6-=100 22(a+bi)(a-bi)=+ab 2 329
1 2 x-= nnnn 22n (1) Divide by the coefficient of the x a==aabab a+bi=+abComplex Modulus 24Łł
2xx-3-=50 n (5) Use Square Root Property aam a+bi=-abi Complex Conjugatenm ( )n (2) Move the constant to the other side. aa
n 32929bb 2 2 x-=–=– xx-=35 (a+bi)(a+bi)=+abi 242n n (3) Take half the coefficient of x, square a=an,if is odd (6) Solve for x it and add it to both sides n n aan,if is even 22 329
3 3929 x=–2xx-3+-=55+-=+= 222244ŁłŁł
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Functions and Graphs Common Algebraic Errors
Constant Function Parabola/Quadratic Function Error Reason/Correct/Justification/Example
22 2 2y==aor fxa x=ay+by+cg (y)=ay++byc ( ) „ 0 and „ 2 Division by zero is undefined!
0 0Grph is a horizontallinepassing
222The graph is a parabola that opens right through the point 0,a . -39=- , -=39 Watch parenthesis! ( ) -„39 ( )
if a > 0 or left if a < 0 and has a vertex
3 3 25 22226xx„ x==xxxx ( ) ( ) bbLine/Linear Function at g--, . aaa 1111aaŁły=mx+bor f (x)=+mxb „+ =„+=2
b+cbc 21+111 Graph is a line with point (0,b) and
Circle 1 A more complex version of the previous --23„+xx slope m. 22 232 error. xx+x-h+y-=kr ( ) ( )
a+ bxabxbxGraph is a circle with radius r and center Slope a + bx =+=+1 „+1 bx aaaaSlope of the line containing the two hk, . ( ) a Beware of incorrect canceling! points xy, and xy, is ( ) ( )11 22
-ax-1 =-+axa ( )Ellipse yy- rise -a (x-1) „--axa 21m== 22 Make sure you distribute the “-“! xx- run (x--h) (yk ) 2 222 22+=1 (x+a)„+xa (x+a)=(x+a)(x+a)=x++2axa Slope – intercept form ab
22 22The equation of the line with slope m Graph is an ellipse with center hk, ( ) x+a„+xa 5= 25=3+4„3+4=3+=47
and y-intercept (0,b) is with vertices a units right/left from the See previous error. x+a„+xa
y=+mxb center and vertices b units up/down from
n More general versions of previous three nnnnnx+a ad x+a„+xa ( )the center. Point – Slope form errors.
The equation of the line with slope m 2222(x+1)=2x+2x+1=2xx++42 ( )Hyperbola and passing through the point xy, is ( )11 22
22 2 22(xx+1)„+( ) x--hyk 2+2=4xx++84 ( ) ( ) ( )y=y+-mxx ( )11 -=1 ab Square first then distribute!
Graph is a hyperbola that opens left and See the previous example. You can not Parabola/Quadratic Function 2+221„+ factor out a constant if there is a power on right, has a center at hk, , vertices a ( ) ( )22 ( )y=ax-h+kfx=ax-+hk( ) () ( ) the parethesis! units left/right of center and asymptotes
1 b 22222-x+a=-+xa 2222 ( )The graph is a parabola that opens up if that pass through center with slope – . -x+a„-+xa aa > 0 or down if a < 0 and has a vertex Now see the previous error.
Hyperbola at hk, . ( ) a22 aaby--kxh( ) ( )„ a 1 a cac-=1 Łł b c === baParabola/Quadratic Function bb 1bbŁłŁł
22 cGraph is a hyperbola that opens up and Łły=ax+bx+cf (x)=ax++bxc ccŁłŁłdown, has a center at hk, , vertices b ( ) aa
units up/down from the center and aThe graph is a parabola that opens up if bbaa 1ŁłŁłasymptotes that pass through center with a > 0 or down if a < 0 and has a vertex === b acŁłc„ c bcbcŁłŁłb bbcbslope – .at --,f . 1ŁłaaaŁłŁł
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
