Логаритъм

[Гост]

Опростете

[Knowledge Greedy]

[tex]\left ( log_ab+log_ba+2 \right )\left ( log_ab+log_{ab}b \right )log_ba-1[/tex]

[tex]=\left ( log_ab+\frac{1}{ log_ab}+2 \right )\left ( log_ab.log_ba+log_{ab}b .log_ba\right )-1=[/tex]

[tex]=\left (\overset{\underbrace{log_ab}}{log_ab}+\frac{1}{ log_ab}+\overset{\underbrace{log_ab}}{2} \right )\left (1+\frac{log_{a}b}{log_{a}ab } .log_ba\right )-1=[/tex]

[tex]=\frac{log^2_ab+2log_ab+1}{log_ab}(1+\frac{1}{log_{a}ab } )-1=[/tex]

[tex]=\frac{\left ( log_ab +1 \right )^2}{ log_ab }. \frac{log_{a}ab+1}{log_{a}ab }-1=[/tex]

[tex]=\frac{\left ( log_ab +1 \right )^2}{ log_ab }. \frac{log_{a}ab+1}{ log_ab +1}-1=[/tex]

[tex]=\frac{\left ( log_ab +1 \right )^{\cancel {2}}}{ log_ab }. \frac{log_{a}ab+1}{\cancel { log_ab +1}}-1=[/tex]

[tex]=\frac{log_a{ab}\left ( log_a{ab} +1 \right ) }{ log_ab}-1=[/tex]

и след привеждане под общ знаменател и съкращаване
[tex]= log_ab + 2log_ba +2[/tex]