4^x-1over2 =2x^2

Answer 1

To solve this equation for x, we can start by finding the common base for both sides of the equation.

First, rewrite 4 as 2^2:

(2^2)^x - 1/2 = 2x^2

Now, use the property (a^m)^n = a^(m*n) to simplify the left side:

2^(2x) - 1/2 = 2x^2

Next, multiply both sides by 2 to get rid of the denominator:

2 * 2^(2x) - 1 = 4x^2

Now, rewrite 2 * 2^2x as (2^1 * 2^2x) using the property a^m * a^n = a^(m + n):

2^(1 + 2x) - 1 = 4x^2

Simplify further:

2^(2x + 1) - 1 = 4x^2

Add 1 to both sides:

2^(2x + 1) = 4x^2 + 1

Now, rewrite the right side as 2^(2) using the property a^(loga(b)) = b:

2^(2x + 1) = 2^(2)

Now, since the bases are the same, we can set the exponents equal to each other:

2x + 1 = 2

Solve for x:

2x = 1

x = 1/2

Therefore, the solution to the equation is x = 1/2.