Question

How do you evaluate ${10^{4{{\log }_{10}}2}}$ ?

Answer 1

In this question we have to find the value of logarithmic function. Logarithm is the inverse function of exponential function. We will use laws of logarithms and laws of exponents to solve the given problem.

Complete step by step answer:
To evaluate the given logarithmic function we will use two laws- law of logarithms and law of exponents. According to the law of logarithmic function of base $10$. We have,
$\Rightarrow {10^{{{\log }_{10}}t}} = t$
Now, according to the law of exponents. We have
${(n)^{ab}} = {({n^a})^b}$
Using the above two formulas. We get,
$\Rightarrow {10^{4{{\log }_{10}}2}} = {({10^{{{\log }_{10}}2}})^4}$
We can write ${10^{{{\log }_{10}}2}} = 2$
So, ${(2)^4} = 16$

Hence, ${10^{4{{\log }_{10}}2}}$ $= 16$.

Note: The logarithm with base $10$ is the most common logarithm in mathematics and also known as decadic logarithm or decimal logarithm named after its base. Logarithm of base $b$ is defined as the power to which the base needs to be raised to yield a given number. For example- using the logarithm of base the logarithm of $100$ equals $2$ because $10$ raised to $2$ equals $100$.