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

nicekikah 2021-09-22 Answered
The value of the logarithmic expression \(\displaystyle{{\log}_{{{2}}}{16}}\)

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Expert Answer

liannemdh
Answered 2021-09-23 Author has 11103 answers
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.
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