Let's simplify the given equation step by step:

16 - 4^x lg(7) = |6 7^x * lg(2) - 24|

Using the property of logarithms: log(a^b) = b * log(a), we can rewrite the equation as:

16 - lg(7^4) = |lg(2^(6 * 7^x)) - 24|

Simplify the logarithmic terms further:

16 - lg(2401) = |lg(2^(6 * 7^x)) - 24|

2401 can be expressed as 7^4:

16 - lg(7^4) = |lg(2^(6 * 7^x)) - 24|

Since lg(7^4) = 4, the equation becomes:

16 - 4 = |lg(2^(6 * 7^x)) - 24|

Simplify the equation further:

12 = |lg(2^(6 * 7^x)) - 24|

To simplify the absolute value expression, consider two cases:

Case 1: lg(2^(6 * 7^x)) - 24 > 0

12 = lg(2^(6 7^x)) - 24
36 = lg(2^(6 7^x))

Convert the logarithmic equation to exponential form:

2^(6 * 7^x) = 10^36

Solve for x in this case.

Case 2: lg(2^(6 * 7^x)) - 24 < 0

12 = -1 (lg(2^(6 7^x)) - 24)
12 = 24 - lg(2^(6 * 7^x))

Also solve for x in this case.

Once you find the solutions for x, substitute them back into the original equation to verify if they satisfy the given equation.