Question

How do you simplify ${{\log }_{20}}\left( {{8000}^{x}} \right)?$

Answer 1

Hint: Now to simplify the given expression we will first use the property of exponents in log. We know that ${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$ hence using this we will get a simplified equation. Now we will use the definition of log to find the value of ${{\log }_{20}}8000$ and hence we will get the simplified expression of the given expression.

Complete step by step solution:
Let us understand the concepts of logarithm.
Now firstly the logarithm of a number is defined on a base value.
Let us say we have an expression ${{\log }_{a}}x$ then a is called the base of logarithm.
Now logarithm gives us the value of exponent. It gives us power to which base must be raised to obtain a certain number.
Hence suppose if we have an equation ${{\log }_{a}}b=n$ then in exponent form we can write it as ${{a}^{n}}=b$ .
Now let us understand the properties of log.
Now if we have multiplication in log then we can expand it as $\log \left( ab \right)=\log a+\log b$ , similarly if we have division in log we can write it as $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ and if we have indices in logarithm then we write it as $\log {{\left( a \right)}^{m}}=m\log a$ .
Now consider the given expression ${{\log }_{20}}\left( {{8000}^{x}} \right)$.
Now by using the property of exponents in log we write the expression as $x{{\log }_{20}}8000$
Now let us find the value of ${{\log }_{20}}8000$.
Let us say that $\Rightarrow {{\log }_{20}}8000=n$
$\begin{align}
  & \Rightarrow {{20}^{n}}=8000 \\
 & \Rightarrow {{20}^{n}}={{20}^{3}} \\
\end{align}$
Hence we get the value of n is 3.
Hence ${{\log }_{20}}8000=3$

Hence we can write $x{{\log }_{20}}8000$ as 3x.

Note: Now consider note that the value of log1 is 0 for all base values. This is because for any real number a we have ${{a}^{0}}=1$ . Also note that if the base of log is not written then we assume the base to be 10. The entry value of a log can never be zero or a negative number.