Math Formulas: Logarithm formulas
Logarithm formulas 1 y = log a x ()ay = x (a;x > 0;a 6= 1) 2 log a 1 = 0 3 log a a = 1 4 log a (mn) = log a m+log a n 5 log a m n = log a m log a n 6 log a m n
Worksheet: Logarithmic Function
Vanier College Sec V Mathematics Department of Mathematics 201-015-50 Worksheet: Logarithmic Function 1 Find the value of y (1) log 5 25 = y (2) log 3 1 = y (3) log 16 4 = y (4) log
Logarithms - Math - The University of Utah
Below is the graph of a logarithm of base a>1 Notice that the graph grows taller, but very slowly, as it moves to the right Below is the graph of a logarithm when the base is between 0 and 1 ***** *** 210 Graphing logarithms Recall that if you know the graph of a function, you can find the graph of
Logarithms and their Properties plus Practice
Definition of a logarithm: If and is a constant , then if and only if In the equation is referred to as the logarithm, is the base , and is the argument The notation is read “the logarithm (or log) base of ” The definition of a logarithm indicates that a logarithm is an exponent is the logarithmic form of
Properties of Exponents and Logarithms
ln x is called the natural logarithm and is used to represent log e x , where the irrational number e 2 : 71828 Therefore, ln x = y if and only if e y = x Most calculators can directly compute logs base 10 and the natural log orF any other base it is necessary to use the change of base formula: log b a = ln a ln b or log 10 a log 10 b
General Logarithms and Exponentials
General exponential functions For a > 0 and x any real number, we de ne ax = ex lna; a > 0: The function ax is called the exponential function with base a Note that ln(ax) = x lna is true for all real numbers x and all a > 0
Linear and Logarithmic Interpolation - CMU
Mar 24, 2004 · This, however, means that the logarithm of these values is on the appropriate linear scale Hence, if the axis in Fig 1 were in fact logarithmic, Eqn 1 would have to be replaced by logx2 ¡logx logx¡logx1 = 1 f ¡1 : (4) Solving this for x, we flnd the logarithmic interpolation formula x = xf 2 x 1¡f 1 (log) : (5) If for instance f = 1
Compound interest, number and natural logarithm
logarithm September 6, 2013 Compound interest, number e and natural logarithm Compound interest If you have money, you may decide to invest it to earn interest The
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Vanier College Sec V Mathematics
Department of Mathematics 201-015-50Worksheet: Logarithmic Function1. Find the value ofy.
(1) log525 =y(2) log31 =y(3) log164 =y(4) log218
=y (5) log51 =y(6) log28 =y(7) log717
=y(8) log319 =y (9) log y32 = 5 (10) log9y=12 (11) log418 =y(12) log9181 =y2. Evaluate.
(1) log31 (2) log44 (3) log773(4)blogb3(3) log2553(4) 16log48
3. Write the following expressions in terms of logs ofx,yandz.
(1) logx2y(2) logx3y2z (3) logpx 3py 2z4(4) logxyz
(5) log xyz (6) logxy 2 (7) log(xy)13 (8) logxpz (9) log 3px 3 pyz (10) log4rx 3y2z4(11) logxrpx
z (12) logrxy 2z 84. Write the following equalities in exponential form.
(1) log381 = 4 (2) log77 = 1 (3) log12
18 = 3 (4) log31 = 0 (5) log 4164=3 (6) log6136 =2 (7) logxy=z(8) logmn=12
5. Write the following equalities in logarithmic form.
(1) 82= 64 (2) 103= 10000 (3) 42=116
(4) 34=181 (5) 12 5 = 32 (6)13 3 = 27 (7)x2z=y(8)px=y6. True or False?
(1) log xy 3 = logx3logy(2) log(ab) = logalogb(3) logxk=klogx (4) (loga)(logb) = log(a+b) (5)logalogb= log(ab) (6) (lna)k=klna (7) log aaa=a(8)ln1x = lnx(9) lnpx xk= 2k7. Solve the following logarithmic equations.
(1) lnx=3 (2) log(3x2) = 2 (3) 2logx= log2 + log(3x4) (4) logx+ log(x1) = log(4x) (5) log3(x+ 25)log3(x1) = 3 (6) log9(x5) + log9(x+ 3) = 1
(7) logx+ log(x3) = 1 (8) log2(x2) + log2(x+ 1) = 28. Prove the following statements.
(1) log pb x= 2logbx(2) log1pb px=logbx(3) logb4x2= logbpx9. Given that log2 =x, log3 =yand log7 =z, express the following expressions
in terms ofx,y, andz. (1) log12 (2) log200 (3) log 143(4) log0:3 (5) log1:5 (6) log10:5 (7) log15 (8) log60007
10. Solve the following equations.
(1) 3 x2 = 12 (2) 31x= 2 (3) 4 x= 5x+1(4) 61x= 10x (5) 32x+1= 2x2(6)101 +ex= 2
(7) 52x5x12 = 0 (8)e2x2ex= 15
11. Draw the graph of each of the following logarithmic functions, and analyze each
of them completely. (1)f(x) = logx(2)f(x) = logx (3)f(x) =log(x3) (4)f(x) =2log3(3x) (5)f(x) =ln(x+ 1) (6)f(x) = 2ln12 (x+ 3) (7)f(x) = ln(2x+ 4) (8)f(x) =2ln(3x+ 6)12. Find the inverse of each of the following functions.
(1)f(x) = log2(x3)5 (2)f(x) = 3log3(x+ 3) + 1 (3)f(x) =2log2(x1) + 2 (4)f(x) =ln(12x) + 1 (5)f(x) = 2x3 (6)f(x) = 233x1 (7)f(x) =5ex+ 2 (8)f(x) = 12e2x13. 15 000$ is invested in an account that yeilds 5% interest per year. After how
many years will the account be worth 91 221.04$ if the interest is compounded yearly?14. 8 000$ is invested in an account that yeilds 6% interest per year. After how
many years will the account be worth 13709.60$ if the interest is compounded monthly?15. Starting at the age of 40, an average man loses 5% of his hair every year. At
what age should an average man expect to have half his hair left?16. A bacteria culture starts with 10 00 bacteria and the number doubles every 40
minutes. (a) Find a formula for the number of bacteria at time t. (b) Find the number of bacteria after one hour. (c) After how many minutes will there be 50 000 bacteria?ANSWERS
1. (1) 2
(2) 0 (3) 12 (4)3 (5) 0 (6) 3 (7)1 (8)2 (9) 2 (10) 13 (11)32 (12)22. (1) 0
(2) 1 (3) 3 (4) 3 (5) 32(6) 643. (1) 2logx+ logy (2) 3logx+ 2logylogz (3) 12 logx+23 logy4logz (4) logx+ logy+ logz (5) logxlogylogz (6) 2logx2logy (7) 13 logx+13 logy (8) logx+12 logz (9) 13 (logxlogylogz) (10) 14 logx+12 logylogz (11) 54
logx12 logz (12) 12 logx+ logy4logz
4. (1) 3
4= 81 (2) 7 1= 7 (3) 12 3 =18 (4) 3 0= 1 (5) 4 3=164 (6) 6 2=136 (7)xz=y (8)m12 =n5. (1) log
864 = 2
(2) log1010000 = 3
(3) log 4116=2 (4) log 3181
=4 (5) log 12 32 =5
(6) log 13 27 =3
(7) log xy= 2z (8) log xy=126. (1) True (2) False (3) True (4) False (5) False (6) False (7) True (8) True