I am stuck solving this problem, any help please..Many thanks
find all numbers x that satisfy the given equation ln(2x)
------- =2
ln(3x)
ln(2x)/ln(3x)=2
ln(2x)=2ln(3x)
ln(2x)=ln(9x²)
raise to power of e
eln(2x)=eln(9x²)
2x=9x²
x(9x-2)=0
x=0, or x=2/9
When working with log equations, always check back to see if the solution results in defined values.
Log(non-positive number) does not exist.
Check x=0:
Both log(2x) and log(3x) give undefined values, and
Lim x->0 log(2x)/log(3x)=1
and not 2 as required.
Check x=2/9:
log(2*x)/log(3*x)
=log(4/9)/log(6/9)
=log(4/9)/log(2/3)
=2log(2/3)/log(2/3)
=2 checks.
Thank you mathmate
To solve the given equation:
ln(2x) / ln(3x) = 2
We need to isolate the variable x. Here's how you can proceed:
Step 1: Rewrite the equation using the properties of logarithms.
ln(2x) = 2 * ln(3x)
Step 2: Apply the property of logarithms that states ln(a^b) = b * ln(a).
ln(2) + ln(x) = 2 * (ln(3) + ln(x))
Step 3: Distribute the 2 on the right side of the equation.
ln(2) + ln(x) = 2 * ln(3) + 2 * ln(x)
Step 4: Move the ln(x) terms to one side of the equation.
ln(x) - 2 * ln(x) = 2 * ln(3) - ln(2)
Step 5: Combine like terms on both sides.
- ln(x) = 2 * ln(3) - ln(2)
Step 6: Simplify the right side.
- ln(x) = ln(3^2) - ln(2)
Step 7: Apply the property of logarithms ln(a) - ln(b) = ln(a / b).
- ln(x) = ln(9 / 2)
Step 8: Exponentiate both sides using the property e^ln(a) = a.
e^(-ln(x)) = e^(ln(9/2))
Step 9: Simplify the left side using the property e^(-ln(a)) = 1 / a.
1 / x = 9 / 2
Step 10: Solve for x by taking the reciprocal of both sides.
x = 2 / 9
Therefore, the solution to the equation ln(2x) / ln(3x) = 2 is x = 2/9.