Answer:
[tex]\displaystyle x=\frac{\log 3}{\log(3)-\log 2}\approx 2.71[/tex]
Step-by-step explanation:
Logarithms
We need to recall these properties of logarithms:
[tex]\log_ax^n=m\log_ax[/tex]
[tex]\log_a(xy)=\log_a(y)+\log_a(y)[/tex]
The equation to solve is:
[tex]3^x=3*2^x[/tex]
Applying logarithms:
[tex]\log(3^x)=\log(3*2^x)[/tex]
Applying the exponent property on the left side and the product property on the right side:
[tex]x\log(3)=\log 3+\log 2^x[/tex]
Applying the exponent property:
[tex]x\log(3)=\log 3+x\log 2[/tex]
Rearranging:
[tex]x\log(3)-x\log 2=\log 3[/tex]
Factoring:
[tex]x(\log(3)-\log 2)=\log 3[/tex]
Solving:
[tex]\boxed{\displaystyle x=\frac{\log 3}{\log(3)-\log 2}}[/tex]
Calculating:
[tex]\mathbf{x\approx 2.71}[/tex]
