Question

Вычислите,используя свойства логарифмов

Answer 1

log₃135=log₃(3³*5)=log₃3³+log₃5=3+log₃5
log₁₅3=1/log₃15=1/(log₃(3*5))=1/(log₃3+log₃5)=1/(1+log₃5)
log₄₀₅3=1/log₃405 = 1/(log₃(3⁴*5))=1/(log₃3⁴+log₃5)=1/(4+log₃5)

$frac{log_3135}{log_{15}3}- frac{log_35}{log_{405}3}= frac{3+log_35}{ frac{1}{1+log_35} }- frac{log_35}{ frac{1}{4+log_35}}=\\=(3+log_35)(1+log_35)-log_35(4+log_35)=\\=3+log_35+3log_35+log_3^25-4log_35-log_3^25=3$