Definition:
$$x = log_b(a) \Leftrightarrow b^x = a \,\,\,\, (for \,\, a,b >0 \,\, and \,\, b<>1)$$
Formula to convert logarithm base g to base b:
$$log_b(x) = log_b(g) * log_g( x )$$
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4^300 = number "a" is a really large number, but we don't care what the actual number is, instead we want to know how many digits the number has:
4^300 = a
300 = log4( a )
and:
log10( a) = log10( 4 ) * log4( a ) = log10( 4 ) * 300
log10( a) = log10( 4 ) * 300
log10( a ) = 180.61799739838872 ( so about 180 digits)
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3^400 = b
400 = log3( b )
and:
log10( b) = log10( 3 ) * log3( b ) = log10( 3 ) * 400
log10( b) = log10( 3 ) * 400
log10( b ) = 190.848501887864976 ( so about 190 digits)
