Question

What is $\log \left( {200} \right)$ ?

Answer 1

Hint: The given problem deals with the use of logarithms. It focuses on the basic definition of the logarithm function and its properties. For such types of questions that require us to simplify logarithmic expressions, we need to have knowledge of all the properties of logarithmic function and applications of each one of them. We should know the base of the logarithmic function is $10$ by default.

Complete step by step answer:
In the given problem, we are required to simplify $\log \left( {200} \right)$. This simplification can be done with the help of logarithmic properties. So, there are various logarithmic properties that can be used to simplify the given logarithmic expression $\log \left( {200} \right)$. Firstly, we need to find the factors of $200$, so that we get a clear idea of which logarithmic property to apply.
So, $200 = 2 \times 2 \times 2 \times 5 \times 5$ .

Now, we also know that the base of logarithmic function is considered as ten by default.
${\log _{10}}\left( {200} \right)={\log _{10}}\left( {2 \times 2 \times 2 \times 5 \times 5} \right)$
Expressing the prime factors in form of exponents and powers, we get,
${\log _{10}}\left( {200} \right)={\log _{10}}\left( {{2^3} \times {5^2}} \right)$
Now, multiplying one $2$ with one $5$ to form the powers of ten, we get,
${\log _{10}}\left( {200} \right)={\log _{10}}\left( {2 \times \left( {2 \times 5} \right) \times \left( {2 \times 5} \right)} \right)$
Computing the product, we get,
${\log _{10}}\left( {200} \right)={\log _{10}}\left( {2 \times 10 \times 10} \right)$
Expressing in powers,
${\log _{10}}\left( {200} \right)={\log _{10}}\left( {2 \times {{10}^2}} \right)$
Now, using the logarithmic property ${\log _z}(x \times y) = {\log _z}x + {\log _z}y$, we get,
\[{\log _{10}}\left( {200} \right)={\log _{10}}\left( 2 \right) + {\log _{10}}\left( {{{10}^2}} \right)\]
Now, using the logarithmic property ${\log _b}({a^n}) = n{\log _b}a$, we get,
\[{\log _{10}}\left( {200} \right)={\log _{10}}\left( 2 \right) + 2{\log _{10}}\left( {10} \right)\]
Now, by basic definition of logarithmic function and understanding of interconversion of logarithmic function to exponential function, we know that ${\log _{10}}(10) = 1$,
\[\therefore {\log _{10}}\left( {200} \right)={\log _{10}}\left( 2 \right) + 2\]

Hence, $\log \left( {200} \right)$ can be simplified as \[\left( {{{\log }_{10}}\left( 2 \right) + 2} \right)\] by the use of logarithmic properties and identities.

Note:The given problem involves use of properties and identities of log function and hence requires us to have a thorough knowledge of the same. We also need to have a basic idea about the applications of the identities and properties in such questions. We should know the basis of logarithm to solve the problem.