Answer:
The answer is 0.
Step-by-step explanation:
To find the result of [tex]log_21\cdot log_32\cdot log_43\cdot log_54\cdot log_65\cdot...\cdot log_{200}199[/tex], we use this logarithm rule:
[tex]\boxed{log_ab=\frac{log_cb}{log_ca} }[/tex]
Therefore, [tex]log_21\cdot log_32\cdot log_43\cdot log_54\cdot log_65\cdot...\cdot log_{200}199[/tex] can be changed into:
[tex]\displaystyle\frac{log_c1}{log_c2} \cdot\frac{log_c2}{log_c3} \cdot\frac{log_c3}{log_c4} \cdot\frac{log_c4}{log_c5} \cdot\frac{log_c5}{log_c6} \cdot...\cdot\frac{log_c199}{log_c200}[/tex]
We can see the pattern that the denominator of a fraction is the same as the numerator of the following fraction. Hence, after we cancel out the "denominator-numerator pairs", it will become:
[tex]\displaystyle\frac{log_c1}{log_c200}[/tex]
[tex]=log_{200}1[/tex]
[tex]=\bf 0[/tex]