Question

How do you simplify \[{\log _{10}}5\]?

Answer 1

Hint: The standard log values of \[{\log _{10}}2,{\log _{10}}3,{\log _{10}}10\] are known. So try to express \[5\] as a product or quotient of any of these numbers. After that use the properties of log that is \[{\log _{10}}\left( {a \cdot b} \right) = {\log _{10}}a + {\log _{10}}b\] or \[{\log _{10}}\left( {\dfrac{a}{b}} \right) = {\log _{10}}a - {\log _{10}}b\].

Complete step by step solution:
To find the value of \[{\log _{10}}5\], first try to express \[5\] as a product or quotient of \[2,3,10\] as the log values of these are known generally.
Now \[5\] can be expressed as \[\dfrac{{10}}{2}\].
\[ \Rightarrow {\log _{10}}5 = {\log _{10}}\left( {\dfrac{{10}}{2}} \right)\]
Now using the property \[{\log _{10}}\left( {\dfrac{a}{b}} \right) = {\log _{10}}a - {\log _{10}}b\],
\[ \Rightarrow {\log _{10}}5 = {\log _{10}}10 - {\log _{10}}2\]
Now \[{\log _{10}}10 = 1\] and \[{\log _{10}}2 = 0.3010\],
\[ \Rightarrow {\log _{10}}5 = 1 - 0.3010\]
\[ \Rightarrow {\log _{10}}5 = 0.6990\]
Hence the value of \[{\log _{10}}5\] is \[0.6990\].

Note: For solving logarithm problems the standard log values of \[{\log _{10}}2,{\log _{10}}3,{\log _{10}}10\], must be memorized. Also one must always try to apply the logarithm properties for solving the problems. Some of the frequently used properties are:
\[{\log _{10}}\left( {a \cdot b} \right) = {\log _{10}}a + {\log _{10}}b\]
\[{\log _{10}}\left( {\dfrac{a}{b}} \right) = {\log _{10}}a - {\log _{10}}b\]
\[{\log _{10}}{a^b} = b{\log _{10}}a\]