120e^(2x)=75e^(3x)Please solve with steps

Step 1. Take natural logarithm to both sides of the equation:
[tex]120e^{2x}=75e^{3x}[/tex]
[tex]ln(120e^{2x})=ln(75e^{3x})[/tex]

Step 2. Apply product rule of logarithms [tex]ln(ab)=ln(a)+ln(b)[/tex]:
[tex]ln(120)+ln(e^{2x})=ln(75)+ln(e^{3x})[/tex]

Step 3. Apply the rule [tex]ln(e^x)=x[/tex]:
[tex]ln(120)+2x=ln(75)+3x[/tex]

Step 4. Solve for [tex]x[/tex]:
[tex]ln(120)+2x=ln(75)+3x[/tex]
[tex]ln(120)-ln(75)=3x-2x[/tex]
[tex]x=ln(120)-ln(75)[/tex]
Apply the quotient rule of logarithms [tex]ln(a)-ln(b)= ln(\frac{a}{b} )[/tex]:
[tex]x=ln( \frac{120}{75} )[/tex]
[tex]x=ln( \frac{8}{5} )[/tex]

We can conclude that the solution of our equation is [tex]x=ln( \frac{8}{5} )[/tex], which is approximately [tex]x=0.47[/tex].