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

Answer:

75

Step-by-step explanation:

Answer 2

Answer:

x=75

Step-by-step 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

$\displaystyle \ln\frac{a}{b}=lna-lnb$

We reduce:

$\displaystyle \ln\frac{4x^2}{10x}=ln30$

Taking off logarithms

$\displaystyle \frac{4x^2}{10x}=30$

Operating

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

$\displaystyle 4x^2=300x$

Dividing by x (assuming x different from 0)

$\displaystyle 4x=300$

Solving

$\boxed{x=75 }$