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

ln(4x)+2ln(x)=1.5
Simplify the left side.
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
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
Solve for x
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