Normally if #a^x=b#, we write #log_a b=x#. Here #a# is the base and is written after the words #"log"# as subscript.
However, when the base is #10#, we just write #log# without using #10# in subscript, which means #log_10b# is written just as #logb# by convention.
Similarly, when base is #e#, we use #ln#, here #n# denotes natural or Napiers log and #lnb=x# means #b=e^x#. Again this is by covention.
When base is different from #10# or #e# we need to mention the base specifically after the word #log#.
Hence #log_aa^x=x# and #lna^x=xlna#
Further #log100^x=log(10^2)^x=log10^(2x)=2x# as #log10=1#
and #ln100^x=ln(10^2)^x=ln10^(2x)=2xln10#
