Download Mth 163 test Ch 3, part 1 Name_______________________Pd

Chapter 3 Study Guide #2
Evaluate:
1. ln e10 =
2. log16 64 =
10
think: e to what power = e
or
ln e10 = x
write in exponent form ex = e10 (one-to-one property)
so x = 10
think: 16 to what power = 64
16x = 64
42x = 43 or
24x = 26
2x = 3
4x = 6
3
x=
2
3. Rewrite the expression as a sum, difference and/or a multiple of logarithms.
log 3
4a5
c
= log3 4a5 – log3 c
= log3 4 + log3 a5 – log3 c
= log3 4 + 5 log3 a – log3 c
log 10r2s7
= log 10 + log r2 + log s7
= log 10 + 2 log r + 7 log s or = 1 + 2 log r + 7 log s
ln
5
xy
1
= ln (xy) 5
1
= ln xy
5
=
1
5
ln x +
1
5
ln y
or
1
5
(ln x + ln y)
4. Rewrite this expression as the logarithm of a single quantity.
log 2 + 4 log x – 2 log y
= log 2 + log x4 - log y2
= log 2x4 - log y2
4
2x
= log
2
y
ln 6 + 2 ln a +
1
ln b
4
1
= ln 6 + ln a2 + ln b 4
= ln 6a2 + ln 4 b
= ln 6a2 4 b
Solve each equation. SHOW YOUR WORK.
x
5. 3 
1
81
6. log x  5
1
3x =
4
3
3x = 3-4
x = -4 (one-to-one property)
7. log 6 36  x
write in exponential form
6x = 36
6x = 62
x = 2 (one-to-one property)
9. ln 2x  4.1
e4.1 = 2x (write in exponential form)
e
write in exponential form
105 = x
100,000 = x
8. 1  3e x  16
3ex = 15 (subtract 1 from both sides)
ex = 5 (divide both sides by 3)
ln 5 = x (write in logarithmic form)
x ≈ 1.609
10. ln x  ln 7  2
ln
x
7
4.1
= x (divide both sides by 2)
2
x ≈ 30.170
11. log (x + 5) = 2
2
10 = x + 5 (write in exponential form)
100 = x + 5
95 = x (add 5 to both sides)
13. log5 25 + log5 (x + 1) = 2
log5 25(x + 1) = 2 (product property)
52 = 25(x + 1) (write in exponential form)
25 = 25x +25
0 = 25x
0=x
e2 =
= 2 (quotient property)
x
(write in exponential form)
7
7e2 = x (multiply both sides by 7)
x ≈ 51.723
12. ln x + ln (x – 3) = ln 10
ln x(x – 3) = ln 10 (product property)
x(x – 3) = 10 (one-to-one property)
x2 – 3x – 10 = 0
(x + 2)(x – 5) = 0
x = -2
x=5
x = -2 is not a solution (domain error)
14. log 104x-2 = 26
1026 = 104x – 2 (write in exponential form)
26 = 4x – 2 (one-to-one property)
28 = 4x
(add 2 to both sides)
7=x
(divide both sides by 4)
15. e 2 x  8e x  48  0
(ex – 4)(ex + 12) = 0
(factor)
e =4
ex = -12
ln 4 = x
ln -12 = x
(write in logarithmic form)
x ≈ 1.386
not a solution, domain error
x