Solve for $x$. Round answers to four decimal places.
$\ln (3 x)+4=7$
Answer (1)
Subtract 4 from both sides: ln ( 3 x ) = 3 .
Exponentiate both sides: 3 x = e 3 .
Divide by 3: x = 3 e 3 .
Approximate to four decimal places: 6.6952 .
Explanation
Isolating the Natural Logarithm We are given the equation ln ( 3 x ) + 4 = 7 and asked to solve for x , rounding the answer to four decimal places. Let's start by isolating the natural logarithm term.
Simplifying the Equation Subtract 4 from both sides of the equation: ln ( 3 x ) + 4 − 4 = 7 − 4 ln ( 3 x ) = 3
Exponentiating Both Sides Now, we exponentiate both sides of the equation using the base e to remove the natural logarithm: e l n ( 3 x ) = e 3
Simplifying the Left Side Using the property e l n ( a ) = a , we simplify the left side of the equation: 3 x = e 3
Solving for x Divide both sides of the equation by 3 to solve for x :
x = 3 e 3
Approximating the Value of x Now, we approximate the value of x to four decimal places. We know that e 3 ≈ 20.0855 , so x = 3 20.0855 ≈ 6.69517 Rounding to four decimal places, we get x ≈ 6.6952 .
Final Answer Therefore, the solution for x rounded to four decimal places is approximately 6.6952 .
Examples
In radioactive decay, the amount of a substance remaining after time t is given by N ( t ) = N 0 e − k t , where N 0 is the initial amount and k is the decay constant. If you know the amount remaining and want to find the time it took to decay to that amount, you would solve an equation involving natural logarithms, similar to the one we just solved. For example, if N ( t ) = 2 1 N 0 , then 2 1 = e − k t , and solving for t involves logarithms.
