Solving the Equation: 4^(x) - 10*2^(x-1) = 24
In this article, we will discuss the solution to the equation 4^(x) - 10*2^(x-1) = 24.
Step 1: Simplify the Equation
First, let's simplify the equation by noticing that 4^(x) can be written as (2^2)^(x) = 2^(2x). Similarly, 2^(x-1) can be written as 2^(2x-2) / 2. So, the equation becomes:
2^(2x) - 10*2^(2x-2) / 2 = 24
Step 2: Multiply by 2 to Eliminate the Fraction
To eliminate the fraction, multiply both sides of the equation by 2:
2^(2x+1) - 10*2^(2x-2) = 48
Step 3: Substitute y = 2x
Let's substitute y = 2x to simplify the equation:
2^y - 10*2^(y-2) = 48
Step 4: Divide by 2^y
Divide both sides of the equation by 2^y:
1 - 10*2^(-2) = 48/2^y
Step 5: Simplify and Solve for y
Simplify the equation:
1 - 10/4 = 48/2^y
1 - 2.5 = 48/2^y
-1.5 = 48/2^y
Now, multiply both sides of the equation by -2^y:
-1.5*2^y = -96
Step 6: Solve for x
Since y = 2x, substitute y = 2x into the equation:
-1.5*2^(2x) = -96
Take the logarithm of both sides with base 2:
(2x) * log2(2) = log2(-96/1.5)
2x = log2(64)
x = log2(64) / 2
x = 3
Therefore, the solution to the equation 4^(x) - 10*2^(x-1) = 24 is x = 3.
