MA 107 Exponentials and Logarithms Worksheet
Solutions
1. Express the following quantity as a single logarithm
22222
22 2 2
2ln( ) ln( 1) ln( 1) ln( ) ln( 1) ln( 1) ln[ ( 1)] ln( 1)
(1) (1)(1)
ln ln ln ( 1)
11
xx x x x x xx x
xx xx x xx
xx
+−−+= +−−+= −−+
⎡⎤⎡ ⎤
−+−
⎡⎤
== =−
⎢⎥⎢ ⎥
⎣⎦
++
⎣⎦⎣ ⎦
2. Find the domain and range of
log( 3)yx=−
3 0 3 So the damain is (3, )xx x−> ⇒ > ∈ ∞
And the range is .
y∈ℜ
3.Find the exact value (no decimal answers) of
33
log(4)log(5)
3+
33 3 3
log (4) log (5) log (4 5) log (20 )
333
+×
===20
Solve for x in problems 4-6
4.
33
log ( 1) log ( 4) 2xx−− + =
2
33 3
11
log ( 1) log ( 4) log 2 3 9
44
37
19( 4)9 36 8 37 8
xx
xx xx
xx x x x
−−
⎛⎞
−− + = = ⇒ = =
⎜⎟
++
⎝⎠
−= + = + ⇒ =− ⇒ =−
Note that the domain of the original equation requires
10 1
40 4
xx 1
x
xx
−> ⇒ > ⎫⇒>
⎬
+> ⇒ >−
⎭.
Obviously, 37
8
x=− is not in the domain, i.e., it is an extraneous solution. Hence the solution does not
exist for this equation.
5.
1
36
xx+=
12
ln3
3 6 ( 1)ln 3 ln 6 (ln 6 ln3) ln 3 ln(2) ln 3 log 3
ln 2
xx xxx x x
+=⇒+ = ⇒ − = ⇒ = ⇒==
6.
.09
4(3) 15 0
x−=
.09 .09 .09 33
15 15 1 15
4(3) 15 0 4(3) 15 (3) 0.09 log log
4 4 0.09 4
xxx
xx
⎛⎞ ⎛⎞
−= ⇒ = ⇒ = ⇒ = ⇒ =
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
7. a. If $3000 is invested into a money market fund at an interest rate of 5.6% compounded quarterly,
how long will it take for the investment to triple?
Compounded quarterly, so =1
nt
r
AP n
⎛⎞
+
⎜⎟
⎝⎠
, where 5.6% 0.056, 4, 3 , rnAPtunkown
=
====
So
()
44
4
0.056 0.056
3 = 1+ 3 1+ 1.014 ln(3) 4 ln 1.014
44
ln(3) 1.0986
19.76
4ln(1.014) 0.0556
tt
t
PP t
t
⎛⎞ ⎛⎞
⇒= = ⇒ =
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⇒= ≈ ≈
