Solve (log_(81)x)((1)/(log_(x)3))=6(1)/(4)

Question

Solve  (log_(81)x)((1)/(log_(x)3))=6(1)/(4)

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Understand given equation: Let's start by understanding the given equation: (log_(81)x)((1)/(log_(x)3))=6(1)/(4). We can use the property of logarithms that states log_(a)b = 1/(log_(b)a), which is known as the change of base formula. This allows us to rewrite the equation in a more familiar form.Apply change of base formula: Applying the change of base formula to the given equation, we get: (log_(81)x)(log_(3)x) = 6(1)/(4) Since 81 is 3 raised to the power of 4 (81 = 3^4), we can rewrite log_(81)x as (1/4)log_(3)x.Substitute and simplify: Substituting log_(81)x with (1/4)log_(3)x in the equation, we have: ((1/4)log_(3)x)(log_(3)x) = 6(1)/(4) Now, let's simplify the left side of the equation by multiplying the two logarithms together.Multiply logarithms: Multiplying the two logarithms together, we get: (1/4)(log_(3)x)^2 = 6(1)/(4) Now, we can multiply both sides of the equation by 4 to get rid of the fraction on the left side.Multiply by 4: Multiplying both sides by 4, we obtain: (log_(3)x)^2 = 6 Now, we can take the square root of both sides to solve for log_(3)x.Take square root: Taking the square root of both sides, we have: log_(3)x = √6 or log_(3)x = -√6 However, since a logarithm cannot have a negative result if the base and the argument are positive and real (which they are in this case), we discard the negative solution.Discard negative solution: We are left with log_(3)x = √6. To find the value of x, we need to rewrite this equation in exponential form, which is x = 3^(√6).

Solve $\left(\log _{81} x\right)\left(\frac{1}{\log _{x} 3}\right)=6 \frac{1}{4}$

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Solve $\left(\log _{81} x\right)\left(\frac{1}{\log _{x} 3}\right)=6 \frac{1}{4}$