Algebra III
Lesson 59
Advanced Logarithm Problems – The
Color of the White House
Advanced Logarithm Problems
Note: Log ( or ln) only works on positive numbers. Loga x = b, for x > 0.
Always double check that this is true with all answers, otherwise throw
the answer out.
Example 59.1
Solve: 3log10 x = log10 16 – log10 2
Use power & quotient rules.
log10 x3 = log10 (16/2)
x3 = 8
x=2
Example 59.2
Solve: log7 (x + 1) + log7 (x – 5) = 1
Use product rule, then rewrite as exponent.
log7 (x + 1)(x – 5) = 1
Check
x2 – 4x – 5 = 71
6 + 1 > 0, 6 – 5 > 0
x2 - 4x – 12 = 0
(x – 6)(x + 2) = 0
x = 6, -2
OK
-2 + 1 > 0, -2 – 5 > 0
So, x = 6
Example 59.3
Solve: 2log3 x – log3 (x – 2) = 2
Use power & quotient rules, then rewrite.
x2
Now, x ≠ 2
= 32
Check
x−2
6 > 0, 6 – 2 > 0
x2 = 9x – 18
x2 – 9x + 18 = 0
3 > 0, 3 – 2 > 0
(x – 6)(x – 3) = 0
So, x = 3,6
x = 6, 3
OK
OK
No OK
The Color of the White House
(2x)/2
(x – 5) + 5
x2
7
x7
=x
Log and exponent are mathematical opposites like these others.
log5 125
=3
10log10 1000
e ln1000
logd d55
log5 125 = log5 53
= 103
= 1000
= 55
= 1000
Example 59.4
Simplify:
42log 42 5
=5
9log3 5
= 32 log3 5 = 3log3 5
Example 59.5
Simplify:
= 3log3 25
2
= 25
Example 59.6
Simplify: 10 4 log10
3
= 10
log10
( 3 )4 = 10
= 10log10 3
2
=9
⎛ 1
log10 ⎜ 3 2
⎜
⎝
⎞
⎟
⎟
⎠
4
Example 59.7
Simplify: 3log3 4 + log 3 5
= 3log3 20
Example 59.8
Simplify:
log e e14
= 14
Example 59.9
Simplify:
log 5 25 = log 5 52
=2
= 20
Practice
a) Solve: log4 (x - 1) + log4 (x + 2) = 1
Use product rule, then rewrite as exponent.
log4 (x - 1)(x + 2) = 1
Check
x2 + x – 2 = 41
2 - 1 > 0, 2 + 2 > 0
x2 + x – 6 = 0
-3 - 1 > 0, -3 + 2 > 0
(x – 2)(x + 3) = 0
So, x = 2
x = 2, -3
b) Simplify:
2log 2 5+ log 2 6
= 2 log 2 30
= 30
OK
No OK
c) Find the area in square meters of the segment bounded by
an arc of measure 120° and the chord which joins the
endpoints of the arc in a circle of radius 100 centimeters.
r
Asegment = Asector - Atriangle
h
θ
r
h = 100sin120
=
= 86.6
1
120
π (100 )2 − (100 )(86.6)
2
360
= 10472 – 4330
= 6142 cm2
6142 cm2
(1 m)2
(100 cm)2
= .6142 m2