To solve this problem we are going to use the power rule of logarithms: [tex]ln(a^{x} )=xln(a)[/tex]. Or in words: the logarithm of a power is the power times the logarithm of the base.
So, the first thing we are going to do is take natural logarithm, [tex]ln[/tex], to both sides of our equation:
[tex]ln(5^{x+2} )=ln(9^{2x+3} )[/tex]
Now, by the power rule of logarithms:
[tex](x+2)ln(5)=(2x+3)ln(9)[/tex]
[tex] \frac{(x+2)ln(5)}{2x+3} =ln(9)[/tex]
[tex] \frac{x+2}{2x+3}= \frac{ln(9)}{ln(5)} [/tex]
[tex] \frac{x+2}{2x+3} =1.365[/tex]
[tex]x+2=1.365(2x+3)[/tex]
[tex]x+2=2.73x+4.095[/tex]
[tex]-1.73x=2.095[/tex]
[tex]x=- \frac{2.095}{1.73} [/tex]
[tex]x=-1.211[/tex]
Using our calculator we can conclude that [tex]x[/tex] is approximately -1.211
