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