# Math Help - Prove this logarithmic eaquation

1. ## Prove this logarithmic eaquation

Hi,
Can anybody prove this quation please?

2. use $log_a(b)=\frac{1}{log_b(a)}$

Spoiler:

$\log_a(N)\log_b(N)+\log_b(N)\log_c(N)+\log_c(N)\lo g_a(N)=\log_a(N)\log_b(N)\log_c(N)[\frac{1}{\log_a(N)}+\frac{1}{\log_b(N)}+\frac{1}{\ log_c(N)}]$

$=\log_a(N)\log_b(N)\log_c(N)[\log_N(a)+\log_N(b)+\log_N(c)]$

$=\log_a(N)\log_b(N)\log_c(N)[\log_N(abc)]=\log_a(N)\log_b(N)\log_c(N)\frac{1}{\log_{abc}(N) }$

3. Here is another way.
$\log_a(N)=\dfrac{\ln(N)}{\ln(a)}$ use that to change all on the left hand side.
You can get LHS to equal $\dfrac{[\ln(N)]^2[\ln(abc)]}{ \ln(a) \ln(b) \ln(c)}$.

Work on the RHS you can reduce it. Equality will follow.