Answer:
x=75
Step-by-step explanation:
Solving Logarithm Equations
The natural logarithm is the inverse function of the exponential function which means
[tex]\displaystyle e^{\ln x}=x[/tex]
We have this equation to solve for x
[tex]2lne^{ln2x}-lne^{10x}=ln30[/tex]
Applying the above property
[tex]2ln2x-ln10x=ln30[/tex]
Also knowing that
[tex]a.lnb=lnb^a[/tex]
We have
[tex]ln(2x)^2-ln10x=ln30[/tex]
Using the fundamental property of logarithms
[tex]\displaystyle \ln\frac{a}{b}=lna-lnb[/tex]
We reduce:
[tex]\displaystyle \ln\frac{4x^2}{10x}=ln30[/tex]
Taking off logarithms
[tex]\displaystyle \frac{4x^2}{10x}=30[/tex]
Operating
[tex]\displaystyle 4x^2=30(10x)[/tex]
[tex]\displaystyle 4x^2=300x[/tex]
Dividing by x (assuming x different from 0)
[tex]\displaystyle 4x=300[/tex]
Solving
[tex]\boxed{x=75 }[/tex]
