Question

If we have an expression as ${{\left( 0.2 \right)}^{x}}=2$ and ${{\log }_{10}}2=0.3010$, then what is the value of $x$.

Answer 1

Hint: To solve this question we have to take logarithm on both side of equation given ${{\left( 0.2 \right)}^{x}}=2$. Then, by applying the logarithm properties and valve given in the question ${{\log }_{10}}2=0.3010$ we obtain a value of $x$. The logarithm properties used to solve this question are as following-
$\log {{\left( m \right)}^{n}}=n\log m$
$\log \dfrac{m}{n}=\log m-\log n$

Complete step-by-step solution:
We have given an equation ${{\left( 0.2 \right)}^{x}}=2$.
We have to find the value of $x$.
Now, we have to take logarithm on both side to solve further, we get
${{\left( 0.2 \right)}^{x}}=2$
$\log {{\left( 0.2 \right)}^{x}}=\log 2$
Now, we know that $\log {{\left( m \right)}^{n}}=n\log m$.
Now, applying to the above equation, we get
$x\log \left( 0.2 \right)=\log 2$
We have given that ${{\log }_{10}}2=0.3010$, when we substitute the value we get
$x\log \left( 0.2 \right)=0.3010$
$x\log \left( \dfrac{2}{10} \right)=0.3010$
Now, we know that $\log \dfrac{m}{n}=\log m-\log n$
Now, applying to above equation, we get
$x\left( \log 2-\log 10 \right)=0.3010$
Now, we put the value of $\log 2$ and $\log 10$ in the above equation, we get
$\begin{align}
  & x\left( 0.3010-1 \right)=0.3010\text{ }\left[ \text{As log10=1} \right] \\
 & x\left( -0.699 \right)=0.3010 \\
\end{align}$
We have to cross multiply to obtain the value of $x$, then value of $x$ will be
\[\begin{align}
  & x=\dfrac{0.3010}{-0.699} \\
 & x=-0.43 \\
\end{align}\]
So, the value of x is $-0.43$.

Note: As given in the question ${{\log }_{10}}2=0.3010$, so we use the base 10 for calculating the value of $\log 10$. There are two logarithms in mathematics, one is a natural logarithm and the other is a common logarithm. The base of the common logarithm is $10$, the base of natural logarithm is $e$ and written as ${{\log }_{e}}x$. Here, $e$ represents a fixed irrational number approximately equal to $2.71828$. When the base of the logarithm is not specified in the question, we generally use a common logarithm.