we can use formula
[tex] log_ab=\frac{ln(b)}{ln(a)} [/tex]
[tex] log_23*log_34*log_45*log_56*log_6 7 *log_7 8=\frac{ln(3)}{ln(2)}*\frac{ln(4)}{ln(3)}*\frac{ln(5)}{ln(4)}*\frac{ln(6)}{ln(5)}*\frac{ln(7)}{ln(6)}*\frac{ln(8)}{ln(7)} [/tex]
now, we can cancel terms
and then we can simplify it
[tex] log_23*log_34*log_45*log_56*log_6 7 *log_7 8=\frac{ln(8)}{ln(2)} [/tex]
[tex] log_23*log_34*log_45*log_56*log_6 7 *log_7 8=\frac{ln(2^3)}{ln(2)} [/tex]
[tex] log_23*log_34*log_45*log_56*log_6 7 *log_7 8=\frac{3ln(2)}{ln(2)} [/tex]
now, we can simplify it
and we get
[tex] log_23*log_34*log_45*log_56*log_6 7 *log_7 8=3 [/tex]...........Answer
