ln(4x)+2ln(x)=1.5

Simplify 2ln(x) by moving 2 inside the logarithm.

ln(4x)+ln(x2)=1.5

Use the product property of logarithms, logb(x)+logb(y)=logb(xy).

ln(4x⋅x2)=1.5

Multiply x by x2 by adding the exponents.

Move x2.

ln(4(x2x))=1.5

Multiply x2 by x.

Raise x to the power of 1.

ln(4(x2x))=1.5

Use the power rule aman=am+n to combine exponents.

ln(4×2+1)=1.5

ln(4×2+1)=1.5

Add 2 and 1.

ln(4×3)=1.5

ln(4×3)=1.5

ln(4×3)=1.5

To solve for x, rewrite the equation using properties of logarithms.

eln(4×3)=e1.5

Exponentiation and log are inverse functions.

4×3=e1.5

Divide each term by 4 and simplify.

Divide each term in 4×3=e1.5 by 4.

4×34=e1.54

Cancel the common factor of 4.

Cancel the common factor.

4×34=e1.54

Divide x3 by 1.

x3=e1.54

x3=e1.54

x3=e1.54

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

x=(e1.54)13

Simplify (e1.54)13.

Apply the product rule to e1.54.

x=(e1.5)13413

Multiply the exponents in (e1.5)13.

Apply the power rule and multiply exponents, (am)n=amn.

x=e1.5(13)413

Cancel the common factor of 3.

Factor 3 out of 1.5.

x=e3(0.5)(13)413

Cancel the common factor.

x=e3⋅(0.5(13))413

Rewrite the expression.

x=e0.5413

x=e0.5413

x=e0.5413

x=e0.5413

x=e0.5413

Verify each of the solutions by substituting them into ln(4x)+2ln(x)=1.5 and solving.

x=e0.5413

The result can be shown in multiple forms.

Exact Form:

x=e0.5413

Decimal Form:

x=1.03862931…

Solve for x natural log of 4x+2 natural log of x=1.5