I keep getting wrong answer for this question.
Write as a single logarithm
log(x+1)+2log(2x+1)-log(x-3)
See previous post.
I got 4x^3+8x^2+5x+1/(x-3)
To write the given expression as a single logarithm, you can use the logarithmic properties. The properties you need to remember are:
1. Product rule: log(base a)(b) + log(base a)(c) = log(base a)(b * c)
2. Quotient rule: log(base a)(b) - log(base a)(c) = log(base a)(b / c)
3. Power rule: n * log(base a)(b) = log(base a)(b^n)
Using these properties, let's simplify the expression step by step:
1. log(x+1) + 2log(2x+1) - log(x-3)
Apply the product rule to the second term:
2 * log(2x+1) = log((2x+1)^2)
2. log(x+1) + log((2x+1)^2) - log(x-3)
Combine the first and second terms using the product rule:
log(x+1) + log((2x+1)^2) = log((x+1) * (2x+1)^2)
3. log((x+1) * (2x+1)^2) - log(x-3)
Apply the quotient rule to the previous term:
log((x+1) * (2x+1)^2) - log(x-3) = log(((x+1) * (2x+1)^2) / (x-3))
So the single logarithm form of the given expression is:
log(((x+1) * (2x+1)^2) / (x-3))
To write the expression as a single logarithm, we can use the properties of logarithms. Here's how to approach it step by step:
Step 1: Apply the Power Rule of Logarithms
The Power Rule of Logarithms states that log(base a)b^c = c * log(base a)b. Using this rule, we can simplify the expression:
log(x+1) + 2log(2x+1) - log(x-3)
= log(x+1) + log(2x+1)^2 - log(x-3)
= log(x+1) + log((2x+1)^2) - log(x-3)
Step 2: Apply the Product Rule of Logarithms
The Product Rule of Logarithms states that log(base a)(b * c) = log(base a)b + log(base a)c. Using this rule, we can further simplify the expression:
= log((x+1) * (2x+1)^2) - log(x-3)
Step 3: Apply the Quotient Rule of Logarithms
The Quotient Rule of Logarithms states that log(base a)(b / c) = log(base a)b - log(base a)c. Using this rule, we can simplify the expression further:
= log(((x+1) * (2x+1)^2) / (x-3))
Therefore, the single logarithm expression equivalent to log(x+1) + 2log(2x+1) - log(x-3) is log(((x+1) * (2x+1)^2) / (x-3)).
Make sure to double-check your simplifications and calculations to avoid any errors.