Question

How do you write \[\log 1000\] in exponential form?

Answer 1

Hint: First find the value of the given logarithmic expression \[\log 1000\] by converting the argument of log, i.e., 1000, into the exponential form having base 10. Use the property of logarithm given as: - \[\log {{a}^{m}}=m\log a\] to simplify the expression. Now, use the identity \[{{\log }_{n}}n=1\] to get the value of log. Once the value of this logarithmic expression is found, use the conversion: - if \[{{\log }_{m}}n=k\] then \[n={{m}^{k}}\] to get the required exponential form.

Complete step-by-step solution:
Here, we have been provided with the logarithmic expression: - \[\log 1000\] and we are asked to convert it into the exponential form. But first we need to find the value of this logarithmic expression.
Now, converting the argument of log, i.e., 1000, into the exponential form, we can write \[1000={{10}^{3}}\], so we get the expression as: -
\[\Rightarrow \log 1000=\log \left( {{10}^{3}} \right)\]
Using the formula: - \[\log {{a}^{m}}=m\log a\], we get,
\[\Rightarrow \log 1000=3\log 10\]
Since, the given log is a common log, i.e., log to the base 10, so we have,
\[\Rightarrow \log 1000=3{{\log }_{10}}10\]
Using the identity: - \[{{\log }_{n}}n=1\], we get,
\[\begin{align}
  & \Rightarrow \log 1000=3\times 1 \\
 & \Rightarrow \log 1000=3 \\
\end{align}\]
Now, we know that if \[{{\log }_{m}}n=k\] the according to the basic definition of logarithm we can write \[n={{m}^{k}}\], so the above relation, we have,
\[\Rightarrow 1000={{10}^{3}}\]
Hence, the above relation shows the exponential form and our answer.

Note: One must know the difference between common log and natural log to solve the above question. Common log has base 10 and natural log has base e. Natural log is denoted by ln. You must remember the basic formulas of logarithm like: - \[\log m+ \log n=\log \left( mn \right)\], \[\log m-\log n=\log \left( \dfrac{m}{n} \right)\], \[\log {{m}^{n}}=n\log m\], \[{{\log }_{\left( {{a}^{b}} \right)}}m=\dfrac{1}{b}{{\log }_{a}}m\]. You must remember the basic definition of the logarithm otherwise it will be difficult to use the conversion rule of logarithmic expression into exponential expression.