$\frac{\left(\log _{x} 5+\log _{5} x-1\right)\left(\log _{x}(5 x)\right)}{1+\left(\log _{x} 5\right)^{3}}=\log _{125} 3$ $\newline$ olduğuna göre, $x^{102}$ değerinin $6$ ile bölümünden kalan kaçtır? $\newline$ A) $1$ $\newline$ B) $2$ $\newline$ C) $3$ $\newline$ D) $4$ $\newline$ E) $5$

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Precalculus

Quotient property of logarithms

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\frac{\left(\log _{x} 5+\log _{5} x-1\right)\left(\log _{x}(5 x)\right)}{1+\left(\log _{x} 5\right)^{3}}=\log _{125} 3

\newline

olduğuna göre,

x^{102}

değerinin

6

ile bölümünden kalan kaçtır?

\newline

1

\newline

2

\newline

3

\newline

4

\newline

5

Simplify Equation: Simplify the given equation. $\newline$ Given: $\frac{(\log_x 5 + \log_5 x - 1)(\log_x (5x))}{1 + (\log_x 5)^3} = \log_{125} 3$ $\newline$ First, let's simplify $\log_x (5x)$ . $\newline$ Using the product rule: $\log_x (5x) = \log_x 5 + \log_x x$ . $\newline$ Since $\log_x x = 1$ , we get $\log_x (5x) = \log_x 5 + 1$ .
Product Rule Application: Substitute $\log_x (5x)$ back into the equation. $\newline$ $\frac{(\log_x 5 + \log_5 x - 1)(\log_x 5 + 1)}{1 + (\log_x 5)^3} = \log_{125} 3$
Substitute Simplified Term: Simplify the numerator. $\newline$ $(\log_x 5 + \log_5 x - 1)(\log_x 5 + 1) = (\log_x 5 + \log_5 x - 1)(\log_x 5) + (\log_x 5 + \log_5 x - 1)$
Numerator Simplification: Simplify further. $\newline$ $(\log_x 5 + \log_5 x - 1)(\log_x 5) + (\log_x 5 + \log_5 x - 1) = (\log_x 5)^2 + \log_x 5 \log_5 x - \log_x 5 + \log_x 5 + \log_5 x - 1$
Combine Like Terms: Combine like terms. $\newline$ $(\log_x 5)^2 + \log_x 5 \log_5 x + \log_5 x - 1$
Substitute Back: Substitute back into the equation. $\newline$ $\frac{(\log_x 5)^2 + \log_x 5 \log_5 x + \log_5 x - 1}{1 + (\log_x 5)^3} = \log_{125} 3$
Simplify Right-hand Side: Simplify the right-hand side. $\newline$ $\log_{125} 3 = \frac{\log 3}{\log 125} = \frac{\log 3}{3 \log 5} = \frac{1}{3} \log_5 3$
Equate Simplified Forms: Equate the simplified forms. $\newline$ $\frac{(\log_x 5)^2 + \log_x 5 \log_5 x + \log_5 x - 1}{1 + (\log_x 5)^3} = \frac{1}{3} \log_5 3$
Solve for x: Solve for $x$ . $\newline$ This step involves solving the equation for $x$ , which is complex and requires logarithmic properties.
Find x^ $102$ mod $6$ : Find $x^{102} \mod 6$ . $\newline$ Assuming $x$ is an integer, we need to find the remainder when $x^{102}$ is divided by $6$ .
Use Modular Arithmetic: Use properties of modular arithmetic. $\newline$ For any integer $x$ , $x^6 \equiv 1 \mod 6$ (by Fermat's Little Theorem). $\newline$ Thus, $x^{102} = (x^6)^{17} \cdot x^0 \equiv 1^{17} \cdot 1 \equiv 1 \mod 6$ .