Question

Given that log(2𝑥+2)

log(𝑥+1)
= 2, find the value of 𝑥
A. -2
B. 2
C. 3
D. 1

Answer 1

[AI-generated answer — Bot]

By using the product rule of logarithms, we can simplify the given expression:

log(2𝑥+2)log(𝑥+1) = 2

log[(2𝑥+2)(𝑥+1)] = 2

(2𝑥+2)(𝑥+1) = 10^2 (taking antilogarithm of both sides)

2(𝑥+1)(𝑥+1) = 100

(𝑥+1)^2 = 50

𝑥+1 = ±√50

𝑥 = -1 ± √50

Out of these two values of 𝑥, only 𝑥 = -1 + √50 is an acceptable solution as plugging in 𝑥 = -1 - √50 in the original equation would result in a negative value inside the logarithm, which is not defined.

Therefore, the answer is 𝑥 = -1 + √50, which is approximately 3.

Answer 2

I think you wanted

log(2x+2) / log(x+1) = 2
That is just a change of base to x+1, giving us
log_x+1_(2x+2) = 2
log_x+1_(2) + log_x+1_(x+1) = 2
log_x+1_(2) = 1
x+1 = 2
x = 1

check: log4/log2 = 2 ok.

Answer 3

[AI-generated answer — Bot]

You are correct. Thank you for pointing out the mistake in the original question. The correct solution is as follows:

log(2𝑥+2)/log(𝑥+1) = 2

log(2𝑥+2) = 2log(𝑥+1)

log(2𝑥+2) = log(𝑥+1)^2

2𝑥+2 = (𝑥+1)^2

2𝑥+2 = 𝑥^2+2𝑥+1

𝑥^2 = 1

𝑥 = ±1

However, 𝑥 = -1 is not an acceptable solution as it would result in a division by zero in the original equation. Therefore, the only acceptable solution is 𝑥 = 1.

Checking the answer:
log(4)/log(2+1) = log(4)/log(3) ≈ 1.2619
log(2(1)+2)/log(1+1) = log(4)/log(2) = 2

The values match, confirming that 𝑥 = 1 is the correct solution. The answer is D.

Answer 4

[AI-generated answer — Step-by-Step Bot]

To solve the equation log(2𝑥+2)log(𝑥+1)=2, we can use the property of logarithms that states that the product of two logs with the same base is equal to the logarithm of the product of their arguments.

So, we can rewrite the equation as log((2𝑥+2)(𝑥+1))=2.

Now, we need to convert the logarithmic equation into an exponential equation. The logarithm with base b is equivalent to b raised to the power of the logarithm.

Therefore, we can rewrite the equation as (2𝑥+2)(𝑥+1)=10^2.

Simplifying the equation, we have (2𝑥+2)(𝑥+1)=100.

Expanding the equation, we have 2𝑥^2+2𝑥+2𝑥+2=100.

Combining like terms, we have 2𝑥^2+4𝑥+2=100.

Subtracting 100 from both sides, we have 2𝑥^2+4𝑥+2-100=0.

Simplifying further, we have 2𝑥^2+4𝑥-98=0.

Dividing the equation by 2, we have 𝑥^2+2𝑥-49=0.

Now, we can solve for 𝑥 using factoring or the quadratic formula.

Factoring the equation, we have (𝑥+7)(𝑥-7)=0.

Setting each factor to zero, we have 𝑥+7=0 and 𝑥-7=0.

Solving for 𝑥, we have 𝑥=-7 and 𝑥=7.

However, we need to check if these values satisfy the original equation log(2𝑥+2)log(𝑥+1)=2.

Substituting 𝑥=-7 into the original equation, we have log(2(-7)+2)log((-7)+1)=log(-12)log(-6) which is not a valid logarithm since it involves taking the logarithm of a negative number.

Therefore, 𝑥=-7 is not a valid solution.

Substituting 𝑥=7 into the original equation, we have log(2(7)+2)log((7)+1)=log(16)log(8) which is not equal to 2.

Therefore, 𝑥=7 is not a valid solution.

Hence, there is no value of 𝑥 that satisfies the equation log(2𝑥+2)log(𝑥+1)=2.

The answer is None of the above.