1. Log10²x+log10x²=log10² 2-1

2. Log4(log2x)+log2(log4x)=2
3. X^logx+5/3= 10^5+log x
4. Log 1/2(x-1)+ log 1/2(x+1)-log1/√2(7-x)=1

Again, having a hard time trying to establish the order of operation.

e.g. #3
I put it into Wolfram the way you typed it , and the webpage interpreted it the same way I did

http://www.wolframalpha.com/input/?i=+X%5Elogx%2B5%2F3%3D+10%5E5%2Blog+x

What a messy equation that would be to solve
I have a feeling you meant something like

X^logx + 5/3= 10^(5+log x)

looks like very messy stuff for the other equations as well, the way you typed it.

To solve the given equations involving logarithms, we will follow some basic rules of logarithms and algebraic techniques.

1. Logarithmic Rule: log(a) + log(b) = log(ab)
Using this rule, we can rewrite the equation as:
log10²x + log10x² = log10²(2-1)

Applying the logarithmic rule on the left side, we get:
log10²(x * x²) = log10²(2-1)

Simplifying further, we have:
log10²(x³) = log10²(1)

Since the bases of the logarithms are the same, we can equate the expressions inside the logarithms:
x³ = 1

Finally, by taking the cube root of both sides, we find:
x = 1

2. In this equation, we have logarithms with different bases. To simplify it, we can use the property of logarithms: log(a) / log(b) = log base b of a.

We can rewrite the equation as:
log4(log2x) + log2(log4x) = 2

Applying the property mentioned earlier, we have:
log(log2x) / log4 + log(log4x) / log2 = 2

Now, simplify each logarithm using the base change formula:
log(log2x) / log(2) / log(4) + log(log4x) / log(4) / log(2) = 2

Since log(x) / log(y) = log base y of x, the equation becomes:
log(log2x) / log2 + log(log4x) / log4 = 2

Combine the logarithms using the property log(a) + log(b) = log(ab):
log(log2x)/log2 * log(log4x) / log4 = 2

Simplifying further, we have:
(log(log2x) * log(log4x)) / (log2 * log4) = 2

Solve this nonlinear equation using numerical methods or approximation techniques to find the value of x.

3. The equation involves both ordinary powers and logarithms. Rearranging the equation, we have:
X^log(x) + 5/3 = 10^5 + log(x)

Since the bases of the exponents and logarithms are different, it is challenging to solve this equation algebraically. We can use numerical methods or approximation techniques to find an approximate solution for x.

4. Similar to the previous equations, this equation involves logarithms with different bases. Using the properties of logarithms, we can simplify the equation as follows:

log(1/2)(x - 1) + log(1/2)(x + 1) - log(1/√2)(7 - x) = 1

Applying the logarithmic rule, we can rewrite the equation as:
log(1/2)((x - 1)(x + 1)) - log(1/√2)(7 - x) = 1

Simplifying further, we have:
log(1/2)((x² - 1)) - log(1/√2)(7 - x) = 1

Applying the logarithmic rule again, we get:
log(1/2)(√2 * (x² - 1)) - log(1/√2)(7 - x) = 1

Since the bases of the logarithms are the same, we can combine the terms:
log(1/2)((√2 * (x² - 1))/(1/√2)(7 - x)) = 1

Simplifying inside the logarithm and multiplying both sides by 2 to eliminate the denominator, we have:
log2(2 * (x² - 1)) / (7 - x) = 2

Finally, we can rewrite the equation as:
log2((2 * x² - 2)/(7 - x)) = 2

Solve this equation using numerical methods or approximation techniques to find the value of x.