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

Question

asked May 4, 2021 32.6k views

2 Answers

Danilo Lemes · Answer 1 · 2021-05-10T02:24:53+0000

Answer:

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

$\boxed{x=75 }$