Question

# Use properties of logarithms to condense the logarithmic expression log

Logarithms
Use properties of logarithms to condense the logarithmic expression log 5 + log 2. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

2021-08-12
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