We have to condense the logarithmic expression as well as we have to find the exact value where expression is:

\(\displaystyle{\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}\)

We know that for general logarithm there is base 10.

So rewriting the given logarithmic expression,

\(\displaystyle{\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}={{\log}_{{{10}}}{\left\lbrace{5}\right\rbrace}}+{{\log}_{{{10}}}{\left\lbrace{2}\right\rbrace}}\)

We know properties of logarithm,

\(\displaystyle{\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}{\log{{\left({a}{b}\right)}}}\)

\(\displaystyle={\log{{\left({a}{b}\right)}}}{1}\)

Applying above property for the given expression, we get

\(\displaystyle{\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}\)

\(\displaystyle={\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}{\log{{\left({a}{b}\right)}}}{\log{{\left({5}\times{2}\right)}}}\)

\(\displaystyle={\log{{\left({10}\right)}}}\)

Hence, condense expression of logarithm is \(\displaystyle{\log{{\left({10}\right)}}}\).

If base of logarithm is 10 then expression value will be

\(\displaystyle{{\log}_{{{10}}}{\left\lbrace{5}\right\rbrace}}+{{\log}_{{{10}}}{\left\lbrace{2}\right\rbrace}}={{\log}_{{{10}}}{\left\lbrace{5}\times{2}\right\rbrace}}\)

\(\displaystyle{{\log}_{{{10}}}{\left\lbrace{10}\right\rbrace}}\)

=1

\(\displaystyle{\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}\)

We know that for general logarithm there is base 10.

So rewriting the given logarithmic expression,

\(\displaystyle{\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}={{\log}_{{{10}}}{\left\lbrace{5}\right\rbrace}}+{{\log}_{{{10}}}{\left\lbrace{2}\right\rbrace}}\)

We know properties of logarithm,

\(\displaystyle{\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}{\log{{\left({a}{b}\right)}}}\)

\(\displaystyle={\log{{\left({a}{b}\right)}}}{1}\)

Applying above property for the given expression, we get

\(\displaystyle{\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}\)

\(\displaystyle={\log{{\left({5}\right)}}}+{\log{{\left({2}\right)}}}{\log{{\left({a}{b}\right)}}}{\log{{\left({5}\times{2}\right)}}}\)

\(\displaystyle={\log{{\left({10}\right)}}}\)

Hence, condense expression of logarithm is \(\displaystyle{\log{{\left({10}\right)}}}\).

If base of logarithm is 10 then expression value will be

\(\displaystyle{{\log}_{{{10}}}{\left\lbrace{5}\right\rbrace}}+{{\log}_{{{10}}}{\left\lbrace{2}\right\rbrace}}={{\log}_{{{10}}}{\left\lbrace{5}\times{2}\right\rbrace}}\)

\(\displaystyle{{\log}_{{{10}}}{\left\lbrace{10}\right\rbrace}}\)

=1