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

Accepted Solution

Step 1. Take natural logarithm to both sides of the equation:

Step 2. Apply product rule of logarithms [tex]ln(ab)=ln(a)+ln(b)[/tex]:

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

Step 4. Solve for [tex]x[/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].