Answer :

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]