e^(x-2)-12^(2x+3)=0
Class is a college level math prep.
Sorry for previous post being incomplete.
I'm thinking:
ln^(x-2)=ln12^(2x+3)
(x-2)ln=(2x+3)ln12
xln-2ln=2xln(12)+3ln(12)
xln-2xln(12)= 2ln+3ln(12)
and not sure
Your notation is incorrect.
"2ln" doesn't mean anything
By definition, ln(e^k) = k
so, let's start over:
e^(x-2) = 12^(2x+3)
take ln of both sides:
x-2 = (2x+3) ln 12
x - 2 = 2ln12 x + 3ln12
x(1 - 2ln12) = 2 + 3ln12
x = (2 + 3ln12)/(1 - 2ln12)
To solve the equation e^(x-2) - 12^(2x+3) = 0, you need to find the value of x that satisfies this equation.
To get started, let's simplify the equation step by step:
1. Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential terms:
ln(e^(x-2) - 12^(2x+3)) = ln(0)
2. Use the logarithm properties to simplify the equation further:
ln(e^(x-2)) = ln(12^(2x+3))
(x-2)ln(e) = (2x+3)ln(12)
Since ln(e) is equal to 1, the equation simplifies to:
(x-2) = (2x+3)ln(12)
Now, we can solve for x:
1. Distribute ln(12) to both terms on the right side of the equation:
x - 2 = 2xln(12) + 3ln(12)
2. Move all terms with x to one side of the equation:
x - 2xln(12) = 3ln(12) + 2
3. Combine like terms:
x(1 - 2ln(12)) = 3ln(12) + 2
4. Divide both sides by (1 - 2ln(12)) to isolate x:
x = (3ln(12) + 2) / (1 - 2ln(12))
This is the solution to the equation e^(x-2) - 12^(2x+3) = 0.