Identify the properties used to perform the following simplifications:
Q1) log(100) = log(400) - log(4)
Q2) log(x^2) = log(x) + log(x)
Q3) log(49) = 2log(7)
Q4) Which properties would you use to simplify the following expression?
log (17x^3) two properties

Properties:
1. log(xy) = logx + logy

2. log(x/y) = logx − logy

3. log(x^r) = r logx

                                                                 Property

Q1) log(100) = log(400) - log(4)   ↔     2.

Q2) log(x²) = log(x) + log(x)           ↔     1.  

Q3) log(49) = 2log(7)                      ↔     3.

Q4) Which properties would you use to simplify the following expression? log (17x³):

Explanation:

Properties

1.  

This property means that the logarithm of a product is equal to the sum of the logariths of the factors.

x² may be written as the product x · x

Then, the logarithm of x² is the same that the logarithm of x · x

And you can apply the rule of the logarithm of a product:

Hence, the property applied to Q2) is the number 1.

2.

This property states that the logarithm of a quotient is equal to the logarithm of the dividend less the logarithm of the divisor.

The number 100 can be written as the quotion 400/4. Then:

And, by the second property:

Thus, the property use to simplify the Q1) is the number 2.

3.

This property states that the logaritm of a power is equal to the exponent multiplied by the logarithm of the base

Then, since 49 = 7²,  you can write:

Therefore, the property used to perform the simplification of Q3) is the third one.

Q4)

The given expression, log (17x³,) is the logarithm of a product: the product of 17 and x³.

Thus, you apply the property number 1:

Now, you can simplify the second term: since x³ is a power you can use the property number 3, logarithm of a power, to find:

Then, the expression simplified is:

b.an increase in a causes b to decrease

72 is the answer to this problem