Question

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

Answer 1

Answer: OPTION B

Step-by-step explanation:

You need to remember the logarithms properties:

$log(a)+log(b)=log(ab)\\\\log(a)-log(b)=log(\frac{a}{b})\\\\log(a)^b=blog(a)$

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

$log(x+3)^2+log(x-7)^3-log(x-2)^5-log(x^2)\\\\log((x+3)^2(x-7)^3)-log(\frac{(x-2)}{(x^2)})\\\\log(\frac{(x+3)^2(x-7)^3}{x^2(x-2)^5})$

This matches with the option B.

Answer 2

Answer:

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]