Find value of x for log1/2(x-1)+log1/2(x+1)-log1/√2(7-x)=1
log1/2(x-1)+log1/2(x+1)-log1/√2(7-x)=1
You seriously need to learn how to use parentheses. Also, the final result will depend on the base of the logarithms. I shall assume base 10.
Assuming you meant
log(1/(2(x-1)))+log(1/(2(x+1)))-log(1/(√2(7-x)))=1
log (1/(2(x-1)) * 1/(2(x+1)) * √2(7-x)) = 1
1/(2(x-1)) * 1/(2(x+1)) * √2(7-x) = 10
I'll let you solve the cubic in any way you like, but somehow I feel the question has been garbled.
I suggest you type in your equation at wolframalpha.com and see how it interprets your text. Then use enough parentheses to make it come out right.
To find the value of x in the equation log1/2(x-1) + log1/2(x+1) - log1/√2(7-x) = 1, we need to simplify the expression using logarithmic properties and then solve for x.
Firstly, let's combine the logarithms using the properties of addition and subtraction:
log1/2(x-1) + log1/2(x+1) - log1/√2(7-x) = 1
log1/2[(x-1)(x+1)] - log1/√2(7-x) = 1
Now, let's simplify the expression inside the logarithm:
log1/2[(x^2 - 1)] - log1/√2(7-x) = 1
Next, we can rewrite the logarithm with a different base using the change of base formula:
log1/2[(x^2 - 1)] = log((x^2 - 1)) / log(1/2)
log1/√2(7-x) = log((7 - x)) / log(1/√2)
Now the equation becomes:
log((x^2 - 1)) / log(1/2) - log((7 - x)) / log(1/√2) = 1
To simplify further, we can solve for each logarithm separately:
log((x^2 - 1)) / log(1/2) = 1 + log((7 - x)) / log(1/√2)
Now, let's solve for each logarithm individually:
log((x^2 - 1)) / log(1/2) = log(2) + log((7 - x)) / log(1/√2)
We know that log(1/2) = log(2) / log(1), and log(1/√2) = log(2) / log(√2), so we can substitute these values:
log((x^2 - 1)) / (log(2) / log(1)) = log(2) + log((7 - x)) / (log(2) / log(√2))
Now we can simplify further:
log((x^2 - 1)) / log(2) = log(2) + log((7 - x)) / (log(2) / (1/2))
Using the properties of logarithms, we can simplify the equation to:
log((x^2 - 1)) = 2log(2) + 2log((7 - x))
Next, let's apply the exponentiation property of logarithms to remove the logarithms:
(x^2 - 1) = 2^2 * 2 * (7 - x)
Simplifying:
x^2 - 1 = 4 * 2 * (7 - x)
x^2 - 1 = 8 * (7 - x)
Expanding:
x^2 - 1 = 56 - 8x
Rearranging:
x^2 + 8x - 57 = 0
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. By factoring or using the quadratic formula, we get the solutions:
x = -9 or x = 6
So, the possible values for x that satisfy the original equation are -9 and 6.