solve the equation 4^2x-2b= (1/8)^x+b for x in terms of b.
1) x=b/7
2) x=7b
3) x=-7b
4)x= -b/7
I will assume that you meant to say ....
4^(2x-2b) = (1/8)^(x+b)
The way you typed it, solving for b would be a total nightmare.
2^(4x - 4b) = 2^(-3x - 3b)
4x - 4b = -3x - 3b
7x = b
x = b/7
(remember 4= 2^2 , and 1/8 = 1/2^3= 2^-3 )
Thank you so much Reiny =]
To solve the equation 4^(2x-2b) = (1/8)^(x+b) for x in terms of b, we can take the logarithm of both sides of the equation and use logarithmic properties to simplify it.
Let's start by taking the logarithm of both sides with base 4. This will get rid of the exponent on the left side of the equation.
log4(4^(2x-2b)) = log4((1/8)^(x+b))
Using the property logb(b^x) = x, we can simplify the left side of the equation to just 2x-2b.
2x-2b = log4((1/8)^(x+b))
Next, we can simplify the right side of the equation using the property logb(a^x) = x*logb(a).
2x-2b = (x+b)*log4(1/8)
Now, we can simplify log4(1/8).
log4(1/8) = log4(1) - log4(8) = 0 - log4(2^3) = -3*log4(2) = -3*(log2(2)/log2(4)) = -3/2
Replacing log4(1/8) with -3/2 in the equation, we have:
2x - 2b = (x + b) * (-3/2)
Distributing (-3/2) to both terms on the right side, we get:
2x - 2b = (-3/2)x - (3/2)b
Now, let's isolate the x terms on one side and the b terms on the other side.
2x - (-3/2)x = - (3/2)b + 2b
Adding the x terms and combining the b terms, we have:
(4/2)x + (3/2)x = (4/2 - 3/2)b
Simplifying both sides, we get:
7x/2 = (1/2)b
Multiplying both sides by 2/7, we finally arrive at the solution for x in terms of b:
x = (1/7)b
Therefore, the correct answer is option 1) x = b/7.