Q.If $\log_{10}(\log_{10}(\log_{10}x))=0$ then $x=\ldots$ (A) $1000$ (B) $10^{10}$ (C) $10$ (D) $0$

The logarithm is defined as the operation that undoes exponentiation: for $a>0,a\ne1$, $y=\log_ax \iff a^y=x$ (for $x>0$) — the logarithmic form and exponential form of the same fact, convertible into each other directly by matching base-to-base and exponent-to-log-value. The domain of $\log_ax$ is $(0,\infty)$ and its range is all of $R$; for the natural base $e$, $\log_ex$ is written $\ln x$.

Three laws let logarithms convert products, quotients and powers into sums, differences and multiples: $\log_a(bc)=\log_ab+\log_ac$ (product rule), $\log_a\frac bc=\log_ab-\log_ac$ (quotient rule), and $\log_ab^c=c\log_ab$ (power rule) — repeatedly misapplying these (e.g. writing $\log(x+y)=\log x+\log y$, which is false) is the single most common logarithm error. A logarithm in one base converts to any other base $b$ via the change of base formula $\log_ax=\dfrac{\log_bx}{\log_ba}$; a special case is the reciprocal identity $\log_ax=\dfrac{1}{\log_xa}$.

Four logarithmic-inequality rules describe order: for base $a>1$, bigger argument means bigger log; for base $0<a<1$, the order flips; and a log is positive exactly when its base and argument sit on the same side of 1, negative when they sit on opposite sides. For a base-10 (common) logarithm, splitting $\log_{10}x=[\log_{10}x]+\{\log_{10}x\}$ into its integer part (the characteristic) and fractional part (the mantissa) gives a shortcut for counting digits: a positive integer with $m$ digits has characteristic exactly $m-1$.