Logarithmic expressions Problems and solutions

Simplifying the logarithmic expressions in mathematics is one type of logarithmic problems. The log expressions are actually simplified for evaluating them. For simplifying the logarithm expressions, we must have knowledge on fundamental operations of mathematics and the properties of the logarithms.

Arithmetic expressions

$(1).\,\,$ Evaluate $\log_{5}{7^{\displaystyle -3\log_{7}{5}}}$

$(2).\,\,$ Evaluate $7\log_{2}{\bigg(\dfrac{16}{15}\bigg)}$ $+$ $5\log_{2}{\bigg(\dfrac{25}{24}\bigg)}$ $+$ $3\log_{2}{\bigg(\dfrac{81}{80}\bigg)}$

$(3).\,\,$ Evaluate $\log_{\sqrt{2}}{\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}}$

$(4).\,\,$ Evaluate $\dfrac{\log_{4}{17}}{\log_{9}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

$(5).\,\,$ Evaluate $\log{2}$ $+$ $16\log{\Big(\dfrac{16}{15}\Big)}$ $+$ $12\log{\Big(\dfrac{25}{24}\Big)}$ $+$ $7\log{\Big(\dfrac{81}{80}\Big)}$