Log2(lgx+2√lgx+1)-2log4(√lgx+1)=1

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To solve this equation, we will first simplify each logarithmic term using the properties of logarithms.

Since log2(x) = log(x) / log(2) and log4(x) = log(x) / log(4), we can rewrite the equation as:

log(lgx + 2√lgx + 1) / log(2) - 2 * log(√lgx + 1) / log(4) = 1

Apply the power rule of logarithms to simplify the equation:

log((lgx + 2√lgx + 1)^2) / log(2) - log((√lgx + 1)^2) / log(4) = 1

Since log(a^b) = b * log(a), we have:

log((lgx + 2√lgx + 1)^2) / log(2) - 2 * log(√lgx + 1) / log(4) = 1

Rewrite the square terms:

2 log(lgx + 2√lgx + 1) / log(2) - 2 log(√lgx + 1) / log(2^2) = 1

Now, combine the logarithms:

2 log(lgx + 2√lgx + 1) / log(2) - 2 log(√lgx + 1) / log(4) = 1

Simplify further:

2 * log(lgx + 2√lgx + 1) / log(2) - log(√lgx + 1)^2 / log(4) = 1

Now, we can apply the logarithmic property log(a^b) = b * log(a) to the equation:

2 log(lgx + 2√lgx + 1) / log(2) - 2 log(√lgx + 1) / 2 * log(2) = 1

Further simplifying:

2 * log(lgx + 2√lgx + 1) / log(2) - log(√lgx + 1) / log(2) = 1

Since log2(x) = log(x) / log(2), we can rewrite the equation as:

2 * log(lgx + 2√lgx + 1) - log(√lgx + 1) = log(2)

Unfortunately, we cannot simplify the above equation further without knowing the exact value of x.