Please clarify my workings for a somewhat complex (for me) logarithm question

Hi all,

I've been confronted with a somewhat complex logarithm question in the form of solving for x. After a long time of thinking and trying various methods, I finally came up with two possible methods for solving the problem. However, I am not sure which (or if either) is correct, or if I have made a mistake along the way. Please clarify the correct method.

The question is:

log2(X ^ log2(x)) = 4

From there, I used the power rule and brought log2(x) down the front of the equation:

log2(x) * log2(x) = 4

Then I added the logs together:

log2(x) + log2(x) = 4

log2(2x) = 4

Then I rearranged the equation to indical form:

2 ^ 4 = 2X

16 = 2X

x = 8

Thus, x = 8. However, I want to suggest an alternate method as well that returns to this part of the equation:

log2(x) * log2(x) = 4

Perhaps instead of adding them together like I did in the above working, I simply multiply the two logs together to achieve:

log2(x ^ 2) = 4

Rearrange and solve to equal:

2 ^ 4 = x ^ 2

16 = x ^ 2

x = 4

Any help will be much appreciated!

Thanks,

Nathaniel