[tex] log_{0.7}( \sqrt{ \frac{2x + 3}{x - 1} } ) = 0[/tex]

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To solve this equation, we first need to rewrite it in a more simplified form before we can apply logarithmic properties.

First, notice that we can rewrite the square root as a fractional exponent:

[tex] log_{0.7}( (\frac{2x + 3}{x - 1})^{1/2} ) = 0[/tex]

Next, we can bring down the exponent using the property of logarithms that states:

[tex] a^{log_a(b)} = b [/tex]

Applying this property, we get:

[tex] log_{0.7}( (\frac{2x + 3}{x - 1})^{1/2} ) = 0 [/tex]
[tex] \frac{2x + 3}{x - 1} = 0.7^0 [/tex]
[tex] \frac{2x + 3}{x - 1} = 1 [/tex]

Now, we can solve for x by cross multiplying:

[tex] 2x + 3 = x - 1 [/tex]
[tex] 2x - x = -1 - 3 [/tex]
[tex] x = -4 [/tex]

Therefore, the solution to the equation is x = -4.