 # WK10 Properties of Logarithmic Anonymous
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Attached is the document of Week 10 Guided Lecture Notes, there are problems about properties of Logarithmic. Complete the document

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Super_Teach12
School: Boston College   Hi, here is your assignment solutions :). Let me know if you need more help ;)

Properties of Logarithms; Logarithmic and Exponential Equations

From last week (5.4)
Post 1
https://mediaplayer.pearsoncmg.com/assets/bIl00_lvtNkCqnHF077MHAI55lbv7h_y
Author in Action: Solve Logarithmic Equations (6:43)
Solving Basic Logarithmic Equations
When solving simple logarithmic equations (they will get more complicated in Section 5.6) follow
these steps:
1. Isolate the logarithm if possible.
2. Change the logarithm to exponential form and use the strategies learned in Section 5.3 to solve
for the unknown variable.
Example 7*: Solve Logarithmic Equations
Solve the following logarithmic equations
(a)* log2 ( 2x +1) = 3
Apply log rule: 𝑎 = 𝑙𝑜𝑔𝑏 (𝑏𝑎 )

3 = 𝑙𝑜𝑔2 (23 ) = 3𝑙𝑜𝑔2 (8)
𝑙𝑜𝑔2 (2𝑥 + 1) = 𝑙𝑜𝑔2 (8)
2𝑥 + 1 = 8
2𝑥 + 1 − 1 = 8 − 1
2𝑥 = 7
2𝑥 7
=
2
2
7
𝑥 = 𝑇𝑟𝑢𝑒
2
𝟕
𝒙=
𝟐

(b)* log x 343 = 3
ln(343)
=3
ln (𝑥 )
ln(343)
𝑖𝑛(𝑥 ) = 3 ln(𝑥 )
ln(𝑥 )

Section 5.5 & 5.6
ln (343) = 3 ln(𝑥 )
3 ln (𝑥 ) = ln (343)
3 ln(𝑥 ) ln (343)
=
3
3
ln(343)
ln (𝑥 ) =
3
ln(𝑥 ) = ln (7)
𝑥 = 7 𝑡𝑟𝑢𝑒
𝒙=𝟕

(c)6 − log(𝑥 ) = 3
6 − 𝑙𝑜𝑔10 (𝑥 ) − 6 = 3 − 6
−𝑙𝑜𝑔10 (𝑥 ) = −3
−𝑙𝑜𝑔10 −3
=
−1
−1
𝑙𝑜𝑔10 (𝑥 ) = 3
Now, apply the log rule:
3 = 𝑙𝑜𝑔10 (103 ) = 𝑙𝑜𝑔10 (1000)
𝑙𝑜𝑔10 (𝑥 ) = 𝑙𝑜𝑔10 (1000)
𝑥 = 1000 𝑡𝑟𝑢𝑒
𝒙 = 𝟏𝟎𝟎𝟎
(d) ln ( x ) = 2
ln(𝑥 ) = ln(𝑒 2 )
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 = 𝑒 2
𝑥 = 𝑒 2 𝑡𝑟𝑢𝑒
𝒙 = 𝒆𝟐

(e) 7 log 6 (4 x) + 5 = −2
7𝑙𝑜𝑔6 (4𝑥 ) + 5 − 5 = −2 − 5
7𝑙𝑜𝑔6 (4𝑥 ) = −7
7𝑙𝑜𝑔6 (4𝑥) −7
=
7
7

Properties of Logarithms; Logarithmic and Exponential Equations
𝑙𝑜𝑔6 (4𝑥 ) = −1
apply log rule:
1
𝑙𝑜𝑔6 (4𝑥 ) = 𝑙𝑜𝑔6 ( )
6
1
4𝑥 =
6
1
1
: 𝑥=
6
24
1
𝑥=
𝑡𝑟𝑢𝑒
24
𝟏
𝒙=
𝟐𝟒

4𝑥 =

(f) log 6 36 = 5 x + 3
5𝑥 + 3 = 𝑙𝑜𝑔6 36
5𝑥 + 3 − 3 = 𝑙𝑜𝑔6 (36) − 3
5𝑥 = −1
5𝑥 −1
=
5
5
𝟏
𝒙=−
𝟓

Steps for solving exponential equations of base e or base 10
1. Isolate the exponential part
2. Change the exponent into a logarithm.
3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the variable.

Example 8*:Using Logarithms to Solve Exponential Equations
Solve each exponential equation.
(a) e x = 7
𝑖𝑛(𝑒 𝑥 ) = 𝑖𝑛 (7)
𝑥𝑙𝑛(𝑒) = ln(7)
𝒙 = 𝒊𝒏 (𝟕)

Section 5.5 & 5.6
(b) (b)* 2e3 x = 6
𝑏 ∗ 2𝑒 3𝑥
6
=
𝑏∗2
𝑏∗2
3
𝑒 3𝑥 =
𝑏
3
𝑖𝑛 (𝑒 3𝑥 ) = 𝑙𝑛 ( )
𝑏
3
3𝑥𝑙𝑛(𝑒) = 𝑙𝑛 ( )
𝑏
3
3𝑥 = 𝑙𝑛 ( )
𝑏
𝟑

𝒙=

𝒍𝒏 (𝒃)
𝟑

(c) (c) e5 x −1 = 9
𝑐𝑒 5𝑥 −1...

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