Solve for x:
{{{ln(x)+x=5}}}
To solve the equation {{{ln(x) + x = 5}}} for x, we will use a combination of algebraic manipulation and the properties of the natural logarithm function.
Step 1: Move the x term to the other side of the equation.
Start by subtracting x from both sides:
{{{ln(x) = 5 - x}}}
Step 2: Exponentiate both sides of the equation.
Since the natural logarithm and the exponential functions are inverse functions, exponentiating both sides of the equation will cancel out the natural logarithm on the left side. We can use the property: e raised to the power of ln(x) is equal to x.
Raise both sides to the power of e:
{{{e^(ln(x)) = e^(5 - x)}}}
Simplifying the left side using the property mentioned above:
{{{x = e^(5 - x)}}
Step 3: Simplify the right side of the equation.
To simplify the right side, we will distribute the exponent to both terms inside the parentheses:
{{{x = e^5 * e^(-x)}}}
Step 4: Solve for x.
To solve for x, we can divide both sides of the equation by e^(-x):
{{{x / e^(-x) = e^5}}}
Step 5: Simplify the left side using the property: e raised to the power of -x is equal to 1 divided by e raised to the power of x.
{{{x * e^x = e^5}}}
At this point, we have an equation involving both x and e^x, which cannot be solved algebraically using elementary functions. Therefore, we would need to use numerical methods such as graphing or iteration methods (e.g., Newton's method) to approximate the value of x.