Solve for X: log4(3x-7)^2 = 10

(3x-7)^2 = 4^10

3x-7 = ±1024
now it's easy

To solve this equation, we will use the properties of logarithms. Before getting started, we need to know that the logarithm function log4(x) can be rewritten as an exponential equation as 4^y = x.

Let's continue with the equation log4(3x-7)^2 = 10.

Step 1: Apply the exponent property of logarithms, which states that if log base a of x equals y, then a raised to the power y equals x. By applying this property to our equation, we get:
4^10 = (3x-7)^2

Step 2: Simplify the right side of the equation:
(3x-7)^2 = 4^10

Step 3: Take the square root of both sides to remove the exponent:
√((3x-7)^2) = √(4^10)

Step 4: Remember that the square root of a number squared is the absolute value of that number:
|3x-7| = 4^5

Step 5: Rewrite 4^5 as a numerical value:
|3x-7| = 1024

Step 6: Split the equation into two cases:
Case 1: 3x-7 = 1024
Case 2: -(3x-7) = 1024 (because |3x-7| is given as an absolute value)

Step 7: Solve Case 1:
3x-7 = 1024
3x = 1024+7
3x = 1031
x = 1031/3

Step 8: Solve Case 2:
-(3x-7) = 1024
-3x + 7 = 1024
-3x = 1024 - 7
-3x = 1017
x = 1017/-3

So the solution to the equation log4(3x-7)^2 = 10 is x = 1031/3 or x = -1017/3.