Solve the equation algebraically. Round the result to three decimal places.
log4 x - log4(x-1) = 1/2
I have no idea how to do this. Please help!
remember log a- log b= log(a/b)
so with that..
log4 (x/(x-1))=1/2
x/(x-1)= 16
then solve for x.
Greetings:
log_4[x] - log_4[x-1] = 1/2 implies log_4[x/(x-1)] = 1/2 which means that 4^(1/2) = x/(x-1) or, 2 = x/(x-1). I shall assume you can finish from there.
Regards,
Rich B.
To solve the given equation algebraically, we can use the properties of logarithms and basic algebraic manipulations. Let's break it down step by step:
1. Start by applying the quotient rule of logarithms: log(a) - log(b) = log(a/b). Rewriting the equation using this rule, we get:
log4(x / (x-1)) = 1/2
2. Convert the equation from logarithmic form to exponential form. Recall that if log(base a) x = c, then a^c = x. Applying this to our equation, we have:
4^(1/2) = x / (x-1)
3. Simplify the left side of the equation: 4^(1/2) is equal to the square root of 4, which is 2. So, we have:
2 = x / (x-1)
4. Next, we can cross-multiply to get rid of the fraction:
2(x - 1) = x
5. Distribute the 2 on the left side of the equation:
2x - 2 = x
6. Move all the x terms to one side of the equation and the constant terms to the other side:
2x - x = 2
x = 2
7. Finally, round the result to three decimal places as requested. In this case, the solution is a whole number, so the rounded result is simply:
x ≈ 2
Therefore, the solution to the equation log4 x - log4(x-1) = 1/2, rounded to three decimal places, is x ≈ 2.