Given:
15. [tex]\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)[/tex]
17. [tex]\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)[/tex]
19. [tex]2^{\log_2100}[/tex]
To find:
The values of the given logarithms by using the properties of logarithms.
Solution:
15. We have,
[tex]\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)[/tex]
Using property of logarithms, we get
[tex]\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)=1[/tex] [tex][\because \log_aa=1][/tex]
Therefore, the value of [tex]\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)[/tex] is 1.
17. We have,
[tex]\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)[/tex]
Using properties of logarithms, we get
[tex]\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)=-\log_{\frac{3}{4}}\left(\dfrac{3}{4}\right)[/tex] [tex][\because \log_a\dfrac{m}{n}=-\log_a\dfrac{n}{m}][/tex]
[tex]\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)=-1[/tex] [tex][\because \log_aa=1][/tex]
Therefore, the value of [tex]\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)[/tex] is -1.
19. We have,
[tex]2^{\log_2100}[/tex]
Using property of logarithms, we get
[tex]2^{\log_2100}=100[/tex] [tex][\because a^{\log_ax}=x][/tex]
Therefore, the value of [tex]2^{\log_2100}[/tex] is 100.
