How to Use logarithm to make simplest linear form of the equations?

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Mathematics

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I missed class due to being sick and have no idea how to do question 4. A-C.

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Assignment 1 1. In a recent survey people were asked if they took a vacation in the summer, winter, or spring in the past year. The results were 73 took a vacation in the summer, 51 took a vacation in the winter, 27 took a vacation in the spring, and 2 had taken no vacation. Also, 10 had taken vacations at all three times, 33 had taken both a summer and a winter vacation, 18 had taken only a winter vacation, and 5 had taken both a summer and spring but not a winter vacation. (1) How many people had been surveyed? (2) How many people had taken vacations at exactly two times of the year? (3) How many people had taken vacations during at most one time of the year? (4) What percentage had taken vacations during both summer and winter but not spring? 2. Shade the region required in for each part. a. ( ∪ ) ∩ b. ( ∩ ∩ ) ∪ ( ∩ ( ∪ )) 3. Simplify the equations below: × . .( × × ) = 1 (solve for y) 4. Use logarithm to make simplest linear form of the following equations: .2 5. 6. 7. 8. =1 .5 =6 Alpha C. Chiang, page14, exercice 2.3, problem: 2 to 4 Alpha C. Chiang, page19, exercice 2.4, problem: 4 to 8 (4 problems) Alpha C. Chiang, page25, exercice 2.5, problem: 5 Alpha C. Chiang, page 47, exercice 3.7, problem: 2 and 3 c. √ =5
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Explanation & Answer

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Question 4
Use logarithm to make simplest linear form of the following equations:
a) 2𝑥 2 𝑦 3 = 1
b) 5𝑥 2 𝑦 = 6𝑥𝑦 2 𝑧
c)

2𝑥 3 𝑦 4
√𝑧

= 5𝑤

Explanation:
Before we go ahead and solve the problems, let us go over some concepts.
An equation is linear if it satisfies the following conditions.
1. The degree of the equation is 1. This mean the highest power of any
variable (e.g. 𝑥, 𝑦, 𝑤, 𝑡, 𝑎, 𝑒𝑡𝑐.) in the equation is less than or equal to 1.
For example: 2 + 3𝑥 = 7 is linear because the highest power of x is 1.
𝑥 3 − 5𝑥 + 2 = 0 is NOT LINEAR because the highest power of the
variable x is 3 which is greater than 1. Do you get the gist.
2. The other condition that a linear equation must satisfy is that the coefficient
of any of the variable wherever it appears in the equation must be a
constant (i.e. a number).
For example in the equation 2 + 3𝑥 = 7, the variable x has a coefficient of
2 which is a constant. 𝑥𝑦 + 7𝑥 2 − 5𝑥 is NOT linear for several reasons.
The first of which is that the coefficient of 𝑥 in the first term is another
variable y (or we can even say y which is also a variable has a coefficient of
x which is not a constant but another variable). The second reason is that the
term 7𝑥 2 has power of the variable x greater than 1 (Remember the first
condition?)

With these conditions in mind it is also important to note that a linear
equation may have more than one variable but the conditions 1 and 2 must
be satisfied. For example 7𝑥 − 2𝑦 + 6 = 𝑧 is a linear equation with three
variables (x, y and z). It is linear because the highest power of any of the
variable is 1 and all the variables have constants (i.e. number values) as
their coefficient.

One more thing before we delve into solving the problems, let us revise our
rules of logarithm.
log 𝑛 (𝑎 ∙ 𝑏) = log 𝑛 𝑎 + log 𝑛 𝑏
𝑎
log 𝑛 (𝑎 ÷ 𝑏) = log ( ) = log 𝑛 𝑎 − log 𝑛 𝑏
𝑏
log 𝑛 𝑎𝑏 = 𝑏 ∙ log 𝑛 𝑎
log 𝑛 𝑛 = 1,

log 𝑛 1 = 0

In log 𝑛 𝑎, n is called the base of the logarithm. If it is omitted, an just
written as log 𝑎, it is assumed that the base is 10 or the natural number 𝑒.
Plea...


Anonymous
Excellent! Definitely coming back for more study materials.

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