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Evaluate the following expression: $$\frac{\left({\mathrm{log}}_{\sqrt[3]{27}}\left(3\right)+{\mathrm{log}}_{\sqrt[49]{5}}\left(25\right)\right)\times \left({\mathrm{log}}_{\sqrt[4]{81}}\left(9\right)-{\mathrm{log}}_{\sqrt[9]{8}}\left(4\right)\right)}{3+{5}^{{\displaystyle \frac{1}{{\mathrm{log}}_{16}\left(25\right)}}}\times {5}^{{\mathrm{log}}_{5}\left(3\right)}}=\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $$\frac{\left({27}^{{\displaystyle \frac{1}{{\mathrm{log}}_{2}\left(3\right)}}}+{5}^{{\mathrm{log}}_{25}\left(49\right)}\right)\times \left({81}^{{\displaystyle \frac{1}{{\mathrm{log}}_{4}\left(9\right)}}}-{8}^{{\mathrm{log}}_{4}\left(9\right)}\right)}{3+{5}^{{\displaystyle \frac{1}{{\mathrm{log}}_{16}\left(25\right)}}}\times {5}^{{\mathrm{log}}_{5}\left(9\right)}}=\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${36}^{{\mathrm{log}}_{6}\left(5\right)}+{10}^{1-{\mathrm{log}}_{10}\left(2\right)}-{3}^{{\mathrm{log}}_{9}\left(36\right)}=\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${\mathrm{log}}_{\sqrt{2}}\left(x\right)+{\mathrm{log}}_{2}\left(x\right)+{\mathrm{log}}_{\sqrt{8}}\left(x\right)=11\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${2}^{2{\mathrm{log}}_{10}\left(4x-1\right)}-{7}^{{\mathrm{log}}_{10}\left(4x\right)}={7}^{{\mathrm{log}}_{10}\left(4x-1\right)}-3\times {4}^{{\mathrm{log}}_{10}\left(4x\right)}\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${x}^{\frac{{\mathrm{log}}_{10}\left(x+5\right)}{3}}={10}^{5+{\mathrm{log}}_{10}\left(x\right)}\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${\left(\sqrt{x}\right)}^{{\mathrm{log}}_{3}\left(x-1\right)}=3\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${\mathrm{log}}_{x}\left(3\right)+{\mathrm{log}}_{3}\left(x\right)={\mathrm{log}}_{\sqrt{x}}\left(3\right)+{\mathrm{log}}_{3}\left(\sqrt{x}\right)+\frac{1}{2}\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${\mathrm{log}}_{4}\left(x+2\right)\times {\mathrm{log}}_{x}\left(2\right)=1\phantom{\rule{0ex}{0ex}}$$ |

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Evaluate the following expression: $${x}^{1+{\mathrm{log}}_{10}\left(x\right)}=10x\phantom{\rule{0ex}{0ex}}$$ |

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Solve the following equation: ${\mathrm{log}}_{10}\left({x}^{2}+19\right)-{\mathrm{log}}_{10}\left(x-8\right)=2$ |