# The value of the logarithmic expression \log_{2}16

The value of the logarithmic expression $$\displaystyle{{\log}_{{{2}}}{16}}$$

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Step 1
Concept Used:
Power property of logarithm $$\displaystyle{{\log}_{{{b}}}{\left({a}^{{{r}}}\right)}}={r}{{\log}_{{{b}}}{\left({a}\right)}}$$
$$\displaystyle{{\log}_{{{b}}}{b}^{{{x}}}}={x}$$
To find the value of the given logarithmic expression first we will rewrite 16 in exponent form as an exponent of base 2 and then use the power property of the logarithm. So we get
$$\displaystyle{{\log}_{{{2}}}{16}}={{\log}_{{{2}}}{2}^{{{4}}}}$$
$$\displaystyle={4}{{\log}_{{{2}}}{\left({2}\right)}}$$ Using power property
Now to simplify further, we use the same base formula $$\displaystyle{{\log}_{{{b}}}{b}^{{{x}}}}={x}$$, to get
$$\displaystyle{{\log}_{{{2}}}{16}}={{\log}_{{{2}}}{2}^{{{4}}}}$$
$$\displaystyle={4}{{\log}_{{{2}}}{\left({2}\right)}}$$ Using power property
$$\displaystyle={4}\cdot{1}$$
$$\displaystyle={4}$$
Hence, the value of the given logarithmic expressionis 4.