Question 162960
Your ability to solve these problems relies on your knowing the rules of logarithms!
"Product rule" {{{Log[b]M + Log[b]N = Log[b](M*N)}}}
"Quotient rule" {{{Log[b]M - Log[b]N = Log[b](M/N)}}}
"Power rule"  {{{p*Log[b]M = Log[b]M^p}}}
Now let's apply these to your problems:
1) {{{2*ln(x) - 5*ln(y)}}} First apply the "power rule" to get:
{{{ln(x^2) - ln(y^5)}}} Now apply the "quotient tule" to get:
{{{highlight(ln(x^2/y^5))}}}
2) {{{Log[3](x^2y)^3}}} Rewrite this by expanding the argument of the logarithm {{{(x^2y)^3 = (x^6*y^3)}}}
{{{Log[3](x^6*y^3)}}} Now apply the "Product rule" to get:
{{{Log[3](x^6) + Log[3](y^3)}}} Now you can apply the "Power rule" to get:
{{{highlight(6*Log[3](x) + 3*Log[3](y))}}}