To exponential digit growth and beyond! Challenge

Given a base \$1 &lt; b &lt; 10\$ and an index \$t \ge 1\$, output term \$x_t\$, defined as follows:


\$x_1 = 11_{10}\$
\$x_{i+1}\$ is obtained by converting \$x_i\$ to base \$b\$ and then reinterpreting its digits in base \$10\$
Output should be in base \$10\$


A walk through for base 5, term 5 would be:


\$x_1 = 11_{10}\$.
\$11_{10} = 21_{5}\$ so \$x_2 = 21_{10}\$.
\$21_{10} = 41_{5}\$ so \$x_3 = 41_{10}\$.
\$41_{10} = 131_{5}\$ so \$x_4 = 131_{10}\$.
\$131_{10} = 1011_{5}\$ so \$x_5 = 1011_{10}\$.
We output the string "1011" or the integer 1011.


Test Cases

Note: these are one indexed

base 2, term 5 --&gt; 1100100111110011010011100010101000011000101001000100011011011010001111011100010000001000010011100011
base 9, term 70 --&gt; 1202167480887
base 8, term 30 --&gt; 4752456545
base 4, term 13 --&gt; 2123103032103331200023103133211223233322200311320011300320320100312133201303003031113021311200322222332322220300332231220022313031200030333132302313012110123012123010113230200132021023101313232010013102221103203031121232122020233303303303211132313213012222331020133


Notes


Standard loopholes are not allowed
Any default I/O method is allowed
You may use different indexes (such as 0-indexed, 1-indexed, 2-indexed, etc) for \$t\$
You may output the first \$t\$ terms.
As this is code-golf, shortest code wins for that language
JavaScript (Node.js), 40 bytes

Thanks to @Neil for saving 5 bytes on this version and 2 bytes on the BigInt version

Takes input as (t)(base), where \$t\$ is 1-indexed.



n=&gt;g=(b,x=11)=&gt;--n?g(b,+x.toString(b)):x


Try it online!



JavaScript (Node.js), 48 bytes (BigInt version)

Takes input as (t)(base), where \$t\$ is 1-indexed. Returns a BigInt.



n=&gt;g=(b,x=11n)=&gt;--n?g(b,BigInt(x.toString(b))):x


Try it online!