Question

How do you evaluate \[\log 5+\log 10+\log 2\]?

Answer 1

Hint: To solve the question we will be using some of the properties of logarithm which are given below. The first property of logarithm states that \[\log a+\log b=\log \left( ab \right)\], here both \[a\And b\] are positive real numbers. The second property states that if any number is not mentioned at the base of the log, assume the base of logarithm to be 10. We should also know another property of logarithm, which states that \[\log {{a}^{n}}=n\log a\]. If the base and argument of logarithm are equal then its value equals 1. We will use these properties to evaluate the given question.


Complete step by step answer:
We are given the expression \[\log 5+\log 10+\log 2\] and we have to evaluate it. As no number is mentioned at the base of the logarithm, we will assume the base to be 10. We know the property of logarithm which states that \[\log a+\log b=\log \left( ab \right)\]. Using this property on the first two terms of the expression we get,
\[\begin{align}
  & \Rightarrow \log 5+\log 10+\log 2 \\
 & \Rightarrow \log \left( 5\times 10 \right)+\log 2 \\
 & \Rightarrow \log \left( 50 \right)+\log 2 \\
\end{align}\]
Again, using the same property on these two terms, we get
\[\begin{align}
  & \Rightarrow \log \left( 50\times 2 \right) \\
 & \Rightarrow \log 100 \\
\end{align}\]
We know that 100 is square of 10, so at the place of 100 in the above equation. We can write \[{{10}^{2}}\]Doing this we get,
\[\begin{align}
  & \Rightarrow \log 100 \\
 & \Rightarrow \log {{10}^{2}} \\
\end{align}\]
We know another property of logarithm, which states that \[\log {{a}^{n}}=n\log a\]. Using this property in the above term we get,
\[\Rightarrow \log {{10}^{2}}=2\times \log 10\]
As the base and argument for \[\log 10\] are the same, \[\log 10=1\]. Substituting this value in the above term we get,
\[\begin{align}
  & \Rightarrow 2\times \log 10=2\times 1 \\
 & \Rightarrow 2 \\
\end{align}\]
Hence the value of the expression \[\log 5+\log 10+\log 2\] is 2.

Note:
 We can evaluate the given expression in one step also. In the solution, we took the first two terms and applied the property on them, we got a new term, and then we again applied the property on this new term and third term. If we want to reduce the number of steps, we can directly apply the property on all three terms at once. This can be done as follows,
\[\begin{align}
  & \log 5+\log 10+\log 2 \\
 & \Rightarrow \log \left( 5\times 10\times 2 \right)=\log \left( 100 \right) \\
 & \Rightarrow \log \left( {{10}^{2}} \right)=2\times \log 10=2\times 1 \\
 & \Rightarrow 2 \\
\end{align}\]