Question

Question #a2865

Answer 1

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)`