Q.Prove that $\log\dfrac{75}{16}-2\log\dfrac{5}{9}+\log\dfrac{32}{243}=\log 2$ OR By the principle of mathematical induction, prove that, for all integers $n\ge 1$, $1^2+2^2+3^2+\cdots+n^2=\dfrac{n(n+1)(2n+1)}{6}$

For a fixed base $0<a\ne1$, the logarithmic function $\log_a(\cdot)$ is defined as the inverse of the exponential $a^x$: $y=a^x\iff\log_ay=x$. Since $a^x$ has domain $R$ and range $(0,\infty)$, $\log_a$ has domain $(0,\infty)$ and range $R$ -- logarithms of non-positive numbers are never defined. Since $a^0=1$ always, $\log_a1=0$ for every base.

The five core laws (all provable directly from the inverse relationship):

Negative-base trick. $\log_{1/a}x=-\log_ax$ (immediate from the change-of-base formula with $\log_a(1/a)=-1$) -- very useful for combining mixed-base logarithmic equations into a single base.

Telescoping chains. A product like $\log_ab\cdot\log_bc\cdot\log_ca$ always collapses to $1$ (change every factor to a common base and watch every intermediate term cancel); more generally $\log_a b_1\cdot\log_{b_1}b_2\cdots\log_{b_{k-1}}b_k=\log_ab_k$. …