Given that $\ln a=5$, express as a simple natural logarithm:

${\log }_a\left(x^2\right)$

Given that $\ln a=5$, express as a simple natural logarithm:

$\frac{1}{2}{\log }_{a}x$

Given that $\ln a=5$, express as a simple natural logarithm:

${\log }_a\left(\frac{x}{3}\right)$

Given that $\ln a=1$, express as a single logarithm:

$\ln \left(x^3\right)+2{\log }_{a}x$

Given that $\ln a=1$, express as a single logarithm:

$2\ln x-3{\log }_{a}x$

Given that $\ln a=1$, express as a single logarithm:

$3{\log }_{a}x+\frac{1}{2}{\log }_{a}x$

Given that $\ln a=1$, express as a single logarithm:

$2\ln \frac{x}{3}-\frac{3}{2}{\log }_{a}x$

Solve the given equation by taking logs of both sides:     [Use $\log 3=0.477121255$ and $\log 5=0.698970004$]

$3^x=25$

Solve the given equation by taking logs of both sides:    [Use $\log 7=0.84509804001426$ and $\log 41=1.6127838567197$]

$7^x=41$