Question

What is the true solution to the equation below? 2 l n e Superscript l n 2 x Baseline minus l n e Superscript l n 10 x Baseline = l n 30 x = 30 x = 75 x = 150 x = 300

Answer 1

75

Explanation:

Answer 2

x=75

Explanation:

Solving Logarithm Equations

The natural logarithm is the inverse function of the exponential function which means

`\displaystyle e^(\ln x)=x`

We have this equation to solve for x

`2lne^(ln2x)-lne^(10x)=ln30`

Applying the above property

`2ln2x-ln10x=ln30`

Also knowing that

`a.lnb=lnb^a`

We have

`ln(2x)^2-ln10x=ln30`

Using the fundamental property of logarithms

`\displaystyle \ln(a)/(b)=lna-lnb`

We reduce:

`\displaystyle \ln(4x^2)/(10x)=ln30`

Taking off logarithms

`\displaystyle (4x^2)/(10x)=30`

Operating

`\displaystyle 4x^2=30(10x)`

`\displaystyle 4x^2=300x`

Dividing by x (assuming x different from 0)

`\displaystyle 4x=300`

Solving

`\boxed{x=75 }`