Rewrite the following expression as a single logarithm. which letter is the correct answer?

OPTION B

Step-by-step explanation:

You need to remember the logarithms properties:

Then, you can rewrite the expression as a single logarithm:

This matches with the option B.

The single logarithm is ㏒[(x + 3)²(x - 7)³/(x²)(x - 2)^5] ⇒ answer B

Step-by-step explanation:

* Lets revise the rule of the logarithmic functions

# ㏒ a + ㏒ b = ㏒ ab

# ㏒ a - ㏒ b = ㏒ a/b

# ㏒ a^n = n ㏒ a

# ㏒ 1 = 0

* Now lets solve the problem

∵ 2㏒(x + 3) + 3㏒(x - 7) - 5㏒(x - 2) - ㏒(x²)

- To make this expression as a single logarithm, change the plus

  to multiplication and minus to division

# 2㏒(x + 3) = ㏒(x + 3)² ⇒ using the third rule up

# 3㏒(x - 7) = ㏒(x - 7)³ ⇒ using the third rule up

# 5㏒(x - 2) = ㏒(x - 2)^5 ⇒ using the third rule up

* lets write the expression

∴ ㏒(x + 3)² + ㏒(x - 7)³- ㏒(x - 2)^5 - log(x²)

# ㏒(x + 3)² + ㏒(x - 7)³  ⇒ change them to single logarithm

∴  ㏒(x + 3)² + ㏒(x - 7)³ = ㏒(x + 3)²(x - 7)³

# - ㏒(x - 2)^5 - log(x²) ⇒ take (-) as a common factor

∴ - (㏒(x - 2)^5 + log(x²)) = - ㏒(x²)(x - 2)^5

∴ ㏒(x + 3)² + ㏒(x - 7)³- ㏒(x - 2)^5 - log(x²) =

  ㏒(x + 3)²(x - 7)³ - ㏒(x²)(x - 2)^5

* Change ㏒(x + 3)²(x - 7)³ - ㏒(x²)(x - 2)^5 to a single logarithm

∴ ㏒(x + 3)²(x - 7)³ - ㏒(x²)(x - 2)^5 = ㏒[(x + 3)²(x - 7)³/(x²)(x - 2)^5]

* The single logarithm is ㏒[(x + 3)²(x - 7)³/(x²)(x - 2)^5]

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

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step-by-step explanation: