Question #a2865

1 Answer
Jun 3, 2015

The important property we need to solve this problem follows from the rule of addition for exponents:
#10^(a+b)=10^a*10^b#

By definition of logarithm,
#log_(10)c# is a number that, if used as an exponent with a base #10#, produces #c#. So, based on this definition, we can write:
#10^(log_(10)c)=c#

If #10^a=p# and #10^b=q#, then, by definition of logarithm, #a=log_(10)p# and #b=log_(10)q#.
Then above rule of addition of exponents can be written as
#10^(log_(10)p+log_(10)q)=p*q# and, also by definition of logarithm,
#log_(10)p+log_(10)q = log_(10)(p*q)#
This is a theorem about addition of logarithms.

Therefore, based on the rule of addition of exponents and the definition of logarithm,
#10^(2*log_(10)x)=10^(log_(10)x+log_(10)x) =10^(log_(10)x)*10^(log_(10)x)=x*x=x^2#

By definition of logarithm mentioned above for #c=x^2#,
#10^(log_(10)(x^2))=x^2=10^(2*log_(10)(x))#

Hence,
#2*log_(10)(x) = log_(10)(x^2)#
which can be directly derived from the theorem about addition of logarithms as
#2*log_(10)(x) = log_(10)(x)+log_(10)(x)=log_(10)(x*x)=log_(10)(x^2)#

Also notice that, since #10^2=100#, the following is true:
#2 = log_(10)(100)#

Therefore, our entire expression can be represented as
#log_(10)(100)+log_(10)(x^2)#

Using the theorem about addition of logarithms, this is equal to
#log_(10)(100*x^2)#