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]