Solve for $x$.
$2 \ln (5 x)=-2$
Do not round any intermediate computations, and round your answer to the nearest tenth.
$x=$
Answer (1)
Divide both sides of the equation by 2: ln ( 5 x ) = − 1 .
Exponentiate both sides using base e : 5 x = e − 1 .
Divide both sides by 5: x = 5 e − 1 .
Simplify: x = 5 e 1 ≈ 0.073576 . Therefore, x = 5 e 1 .
Explanation
Problem Analysis We are given the equation 2 ln ( 5 x ) = − 2 and we need to solve for x .
Divide by 2 First, divide both sides of the equation by 2: 2 2 ln ( 5 x ) = 2 − 2 ln ( 5 x ) = − 1
Exponentiate Next, we exponentiate both sides of the equation using the base e to remove the natural logarithm: e l n ( 5 x ) = e − 1 Since e l n ( a ) = a , we have: 5 x = e − 1
Isolate x Now, divide both sides by 5 to isolate x :
5 5 x = 5 e − 1 x = 5 e − 1
Simplify Finally, we can rewrite e − 1 as e 1 , so: x = 5 e 1 We can approximate this value as: x ≈ 5 × 2.71828 1 ≈ 13.5914 1 ≈ 0.073576 Rounding to the nearest thousandth, we get x ≈ 0.074 .
Final Answer Therefore, the solution for x is: x = 5 e 1 ≈ 0.073576 Rounding to the nearest thousandth, we have x ≈ 0.074 .
Examples
Imagine you are calculating the decay of a radioactive substance. The amount of the 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 need to solve an equation involving the natural logarithm, similar to the one in this problem. This type of problem arises in various fields, including physics, chemistry, and finance, where exponential growth or decay is involved. Understanding how to solve logarithmic equations is crucial for modeling and analyzing these phenomena.
