Question

How do you simplify ${{10}^{{{\log }_{10}}\left( 19 \right)}}$ ?

Answer 1

Hint: For answering this question we have been asked to simplify the given logarithmic expression and reduce it as much as possible. The given expression is ${{10}^{{{\log }_{10}}\left( 19 \right)}}$ which is in the form of ${{a}^{{{\log }_{a}}\left( b \right)}}$ by seeing this we remember the formulae ${{a}^{{{\log }_{a}}\left( b \right)}}=b$ . We will use this formula in this question.

Complete step by step answer:
Now considering from the question we have been asked to simplify the given expression is ${{10}^{{{\log }_{10}}\left( 19 \right)}}$.
We can conclude by observing the given expression in the question that it is in the form of ${{a}^{{{\log }_{a}}\left( b \right)}}$.
From the basics we are aware of the formulae ${{a}^{{{\log }_{a}}\left( b \right)}}=b$. As the given expression is in the similar form we will use this logarithmic formula to answer this question.
This formula can be obtained from the basic definition of logarithm which is mathematically given as ${{a}^{x}}=y\Rightarrow {{\log }_{a}}y=x$ .
Let us assume ${{\log }_{a}}b=u$ then we can say that $\Rightarrow {{a}^{u}}=b$ .
If we replace ${{\log }_{a}}b$ with $u$ in the general form of the given expression ${{a}^{{{\log }_{a}}\left( b \right)}}$ we will have $\Rightarrow {{a}^{{{\log }_{a}}\left( b \right)}}={{a}^{u}}$
As we know that previously that the value of ${{a}^{u}}$ is $b$ hence it is proved that ${{a}^{{{\log }_{a}}\left( b \right)}}=b$ .
By using this formula in the expression given in this question we will have ${{10}^{{{\log }_{10}}\left( 19 \right)}}=19$ . Hence the given expression is simplified.

Note: For solving this type of questions more efficiently we need practice more and more questions. Similarly we have many more logarithmic formulae like $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ and many more.