A natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted by ln(x) and is defined as the power to which the number e (approximately 2.718) must be raised to equal a given number.
The main difference between a natural logarithm (ln) and a common logarithm (log) is the base. A natural logarithm has a base of e, while a common logarithm has a base of 10. In other words, ln(x) represents the power to which e must be raised to equal x, while log(x) represents the power to which 10 must be raised to equal x.
To solve a natural logarithm equation, you can use the properties of logarithms to rewrite the equation in a simpler form. Then, you can use algebraic methods to isolate the logarithm and solve for the variable. In the case of lnx+ ln(x-1) = 1, you can combine the two logarithms using the product rule to get ln(x(x-1)) = 1, and then use the definition of a logarithm to get x(x-1) = e. Finally, you can use the quadratic formula to solve for x.
The possible solutions to this equation will depend on the restrictions of the natural logarithm. Since the domain of ln(x) is x > 0, and the domain of ln(x-1) is x-1 > 0, the possible solutions for this equation are x = 2 and x = e+1.
To check if a solution to a natural logarithm equation is valid, you can substitute the solution back into the original equation and see if it satisfies the equation. In the case of lnx+ ln(x-1) = 1, you can substitute x = 2 and x = e+1 to see if the equation holds true. If it does, then the solutions are valid.