I'm not sure how to calculate the first googol of digits of log10(e), but here's the first 150(approximately) of them...

0.43429448190325182765112891891660508229439700580366656611445378316586464920 8870774729224949338431748318706106744766303733641679287158963906569221064663

We need to deal with the digits immediately after the decimal point in googol*log10(e), which are .187061

frac__[log(googol!)__?] = frac__[halflog2pi + 50 + googol(100-log10(e))__?]

= frac{halflog2pi + frac

[googol(100-log10(e))?]} = frac[.399090 + (1- .187061)?] = .21202910 ** .212029 = 1.629405

Which means that googol! starts with 1629