Convert between a multi-term sum of logarithms and a power Algebra I

We know that:

m\log_ax=\log_ax^{m}

So, we have:

6\log_3\left(x\right)+4\log_3\left(y\right)-2\log_3\left(z\right)=\log_3\left(x^{6}\right)+\log_3\left(y^{4}\right)-\log_3\left(z^{2}\right)

We know that:

\log_a\left(x\right)+\log_a\left(y\right)=\log_a\left(x\cdot y\right)

And:

\log_a\left(x\right)-\log_a\left(y\right)=\log_a\left( \dfrac{x}{y} \right)

So, we have:

\log_3\left(x^{6}\right)+\log_3\left(y^{4}\right)-\log_3\left(z^{2}\right)=\log_3\left( \dfrac{x^{6}y^{4}}{z^{2}} \right)

We know that:

m\log_ax=\log_ax^{m}

So, we have:

2\log\left(a\right)-3\log\left(b\right)+4\log\left(c\right)=\log\left(a^{2}\right)-\log\left(b^{3}\right)+\log\left(c^{4}\right)

We know that:

\log_a\left(x\right)+\log_a\left(y\right)=\log_a\left(x\cdot y\right)

And:

\log_a\left(x\right)-\log_a\left(y\right)=\log_a\left( \dfrac{x}{y} \right)

So, we have:

\log\left(a^2\right)-\log\left(b^3\right)+\log\left(c^4\right)=\log\left(\dfrac{a^{2}c^{4}}{b^{3}}\right)

We know that:

m\log_ax=\log_ax^{m}

So, we have:

4\ln\left(x\right)-3\ln\left(y\right)=\ln\left(x^{4}\right)-\ln\left(y^{3}\right)

We know that:

\log_a\left(x\right)-\log_a\left(y\right)=\log_a\left( \dfrac{x}{y} \right)

So, we have:

\ln\left(x^4\right)-\ln\left(y^3\right)=\ln\left(\dfrac{x^{4}}{y^{3}}\right)

We know that:

m\log_ax=\log_ax^{m}

So, we have:

3\log_5\left(m\right)+7\log_5\left(n\right)=\log_5\left(m^{3}\right)+\log_5\left(n^{7}\right)

We know that:

\log_a\left(x\right)+\log_a\left(y\right)=\log_a\left(x\cdot y\right)

So, we have:

\log_5\left(m^3\right)+\log_5\left(n^7\right)=\log_5\left( m^{3}n^{7} \right)

We know that:

\log_a\left(x\cdot y\right)=\log_a\left(x\right)+\log_a\left(y\right)

So, we have:

\log_2\left( x^{5}y^{6} \right)=\log_2\left( x^{5} \right)+\log_2\left( y^{6} \right)

We know that:

\log_ax^{m}=m\log_ax

So, we have:

\log_2\left( x^{5} \right)+\log_2\left( y^{6} \right)=5\log_2x+6\log_2y

We know that:

\log_a\left(x\cdot y\right)=\log_a\left(x\right)+\log_a\left(y\right)

So, we have:

\ln\left(m^{2}n^{4}\right)=\ln\left(m^{2}\right)+\ln\left(n^{4}\right)

We know that:

\log_ax^{m}=m\log_ax

So, we have:

\ln\left(m^{2}\right)+\ln\left(n^{4}\right)=2\ln\left(m\right)+4\ln\left(n\right)

We know that:

\log_a\left(\dfrac{x}{y}\right)=\log_a\left(x\right)+\log_a\left(y\right)

So, we have:

\log\left(\dfrac{x^{5}}{y^{7}}\right)=\log\left(x^{5}\right)-\log\left(y^{7}\right)

We know that:

\log_ax^{m}=m\log_ax

So, we have:

\log\left(x^{5}\right)-\log\left(y^{7}\right)=5\log\left(x\right)-7\log\left(y\right)