Properties of Logarithms
Power Property
(bx)y = bxy
Quotient Property
bx = bx­y
by
Product Property
bx * by = bx+y
Properties of Logarithms
If b, x, and y are positive numbers, b ≠ 1 and p is a real number, then:
1. Product Property: logb xy = logbx + logby
2. Quotient Property: logb x/y = logbx ­ logby
3. Power Property: logbxp = plogbx
ExA. Using the Properties of Logarithms, express each problem in terms of ln2 and ln3.
1.) ln54
2)
ln(9/8)
3)
log(22 * 33)
log22 + log33
2log2 + 3log3
ln(32/23)
ln32 ­ ln23
2ln3 ­ 3ln2
ln (2*33)
ln2 + ln33
ln2 + 3ln3
log108
ExB. Express each logarithm in terms of log5 and log3
1)
log75
log(52 * 3)
log52 + log 3
2log5 + log3
2)
log5.4
log27/5
log33/5
log33 ­ log5
3log3 ­ log5
3)
log81/125
log34/53
log34 ­ log53
4log3 ­ 3log5
1
ExC. Express each logarithm in terms of log 2 and log 3
1)
log96
2)
log 32/9
ExD. Use the properties to simplify/evaluate logarithms. Remember: to bring #'s out of you rewrite them using fractional exponents.
1) log4 √
5 64
3
log4√
5 4
log4 43/5
3/5(log44)
3/5(1)
3/5
2) 5 ln e2 ­ ln e3
3) log6 √336
4) ln e9 + 4 ln e3
5(2 ln e) ­ 3 ln e
10 ln e ­ 3 ln e
10(1) ­ 3(1)
10 ­ 3
7
log6 √362
log6 62/3
2/3(log6 6)
2/3(1)
2/3
9 ln e + 4(3 ln e)
9 ln e + 12 ln e
9(1) + 12(1)
9 + 12
21
5) log2∛32
log2∛25
log2 25/3
5/3 log22
5/3(1)
5/3
6) 3 ln e4 ­ 2 ln e2
3(4ln e) ­ 2(2ln e)
12 ln e ­ 4 ln e
12(1) ­ 4(1)
12 ­ 4
8
Use the properties of logarithms to expand each expression.
1) log 12x5y­2
log12 + log x5 + log y­2
log12 + 5log x + log 1/y2
log 12 + 5log x + log 1 ­ log y2
log 12 + 5log x + log 1 ­ 2 log y
log 12 + 5log x + 0 ­ 2 log y
log 12 + 5log x ­ 2 log y
2) ln x2 √(4x + 1)
ln x2 ­ ln√(4x + 1) 2ln x ­ ln (4x + 1)½
2ln x ­ ½ln(4x + 1)
2
Use the properties of logarithms to expand each expression.
3) log13 6a3bc4
4) ln 3y + 2 4∛y
log136 + log13a3 + log13b + log13c4
log136 + 3log13 a + log13 b + 4log13 c
ln (3y + 2) / 4y1/3
ln(3y + 2) ­ ln 4y1/3
ln(3y + 2) ­ (ln 4 + ln y1/3)
ln(3y + 2) ­ (ln 4 + 1/3ln y)
ln(3y + 2) ­ ln 4 ­ 1/3ln y
Use the properties of logarithms to condense each expression
2) 6ln(x ­ 4) + 3ln x
1.) 4 log3 x ­ 1/3 log3 (x + 6)
ln(x ­ 4)6 + ln x3
ln(x ­ 4)6 * x3
ln x3(x­4)6
log3 x4 ­ log3(x + 6)1/3
log3 x4 / log3(x+6)1/3
log3 x4 (x + 6)1/3
log3 x4 ∛(x + 6)
4) 5 ln(x+1) + 6ln x
3) ½log4 x ­ 3 log4(x­2)
log4 x½ ­ log4(x ­ 2)3
ln(x+1)5 + ln x6
log4 x½
(x ­ 2)3
ln (x+1)5 * x6
ln x6(x+1)5
log4 √x (x ­ 2)3
Use a change of base formula to evaluate each expression
Ex log3 5 = log5/log3 ≈ 1.465
OR log3 5 = ln5/ln3 ≈ 1.465
a) log½ 6 = log 6/log½
b) log78 4212 = ln4212/ln78
c) log15 33 = log33/log15
d) log1/3 10 = ln10/ln(1/3)
3