Let$MULT=\{a\#b\#c|a,b,c$are binary natural numbers and $a\times b=c\}$. Showthat $MULT\in L$.

The class L: L is the class of languages that are decidable in the logarithmic space on the deterministic truing machine. It shows L = SPACE (log n).

Let M be the DTM that decides the MULT in logarithmic space. decides,

The construction of M is,

$M="Oninputa\#b\#c:$

1. Reject if any of the three strings aren't a binary number as stated above.

2. Set up a binary counter named i those points to the letter O.

3. Set the maximum$max(0,i+1-lengthof1)$for a binary counter j.

4. Set a binary counter from k to i - j to start.

5. Now add $x\leftarrow x+a\left[j\right]*b\left[k\right]$to the binary counter.

6. Repeat steps 3–5 minutes $(i,n-1)$times.

7. Calculate $x=floor\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$by repeating steps 2n -1 times.

Thus, by clearing the that runs into the log space and decides MULT.

So, it proves the condition as MULT$\in L$.