solve for x given that (1/8)^12x =2^x+3
I think you meant
(1/8)^(12x) =2^(x+3)
(2^-3)^(12x) = 2^(x+3)
2^(-36x) = 2^(x+3)
then
-36x = x+3
-37x = 3
x = -3/37
To solve for x in the equation (1/8)^(12x) = 2^(x+3), we need to simplify both sides of the equation and then apply logarithms. Here's the step-by-step process:
Step 1: Simplify the equation.
We can rewrite (1/8)^(12x) as (2^(-3))^(12x) since 1/8 is equivalent to 2^(-3).
Therefore, the equation becomes 2^(-36x) = 2^(x+3).
Step 2: Apply logarithms.
Since the bases on both sides of the equation are equal (both are 2), we can apply logarithms to both sides. The most commonly used logarithm is the natural logarithm (log base e) or the common logarithm (log base 10).
Let's use the natural logarithm (ln) in this case.
Applying the logarithm to both sides gives us:
ln(2^(-36x)) = ln(2^(x+3)).
Using the exponent rule of logarithms, we can write this as:
-36x * ln(2) = (x+3) * ln(2).
Step 3: Resolve the equation.
Distribute ln(2) on the right side to get:
-36x * ln(2) = x * ln(2) + 3 * ln(2).
Now, move all terms with x to one side and the constant terms to the other side:
-36x * ln(2) - x * ln(2) = 3 * ln(2).
Factor out x:
(-36ln(2) - ln(2)) * x = 3 * ln(2).
Combine like terms:
-37ln(2) * x = 3 * ln(2).
Step 4: Solve for x.
Divide both sides of the equation by (-37ln(2)):
x = (3 * ln(2)) / (-37ln(2)).
Simplify:
x = -3/37.
So, x is equal to -3/37.