lnx^5 - lnx^2 = 3 solve for the unknowns as a list.
I tried using my calculator but apparently my answer is opposite ends of the universe
since log x^n = n log x, we have
5lnx - 2lnx = 3
3lnx = 3
lnx = 1
x = e
don't know what list you might be involving
To solve the equation lnx^5 - lnx^2 = 3, you can use logarithmic properties and algebraic techniques. Let's go through the steps together:
Step 1: Simplify the equation using logarithmic properties.
Using the property of logarithms, ln(a) - ln(b) = ln(a/b), we can simplify the equation:
ln(x^5) - ln(x^2) = ln(x^5 / x^2) = ln(x^3)
So, our equation becomes ln(x^3) = 3.
Step 2: Exponentiate both sides of the equation.
To eliminate the natural logarithm, we can exponentiate both sides of the equation by raising e to the power of both sides. The exponential function "e^x" and the natural logarithm function "ln(x)" are inverse functions, so they cancel each other out.
e^(ln(x^3)) = e^3
On the left side, the exponential of the natural logarithm will simply return the original argument:
x^3 = e^3
Step 3: Solve for x by taking the cube root of both sides.
To isolate x, we can take the cube root of both sides:
x = (e^3)^(1/3)
Simplifying the right side further:
x = e^(3/3)
Since 3/3 equals 1, we have:
x = e^1
Therefore, the final solution for x is:
x = e
The solution for x is the Euler's number, approximately 2.71828.
So, the solution to the equation lnx^5 - lnx^2 = 3 is x = e, approximately 2.71828.
Note: The Euler's number (e) is a mathematical constant and can be estimated using a calculator or mathematical software.