What is the true solution to $2 \ln e^{\ln 5 x}=2 \ln 15 ?$
A. $x=0$
B. $x=3$
C. $x=9$
D. $x=15$
Answer (1)
Simplify the equation using the property ln e u = u .
Divide both sides of the equation by 2.
Equate the arguments of the logarithms.
Solve for x: x = 5 15 = 3 . The true solution is 3 .
Explanation
Understanding the Problem We are given the equation $2
ln e^{\ln 5 x}=2
ln 15 an d a s k e d t o f in d t h e t r u eso l u t i o n f ro m t h eo pt i o n s x=0 , x=3 , x=9 , x=15$.
Simplifying the Equation First, we simplify the left side of the equation using the property that ln e u = u . Therefore, ln e l n 5 x = ln 5 x . Substituting this into the original equation, we have
2 ln 5 x = 2 ln 15
Dividing by 2 Next, we divide both sides of the equation by 2:
ln 5 x = ln 15
Equating Arguments Since the natural logarithms are equal, their arguments must be equal. Thus,
5 x = 15
Solving for x Now, we solve for x by dividing both sides of the equation by 5:
x = 5 15
x = 3
Final Answer Therefore, the true solution to the equation is x = 3 .
Examples
Imagine you are trying to determine the growth rate of a population or the decay rate of a radioactive substance. Logarithmic equations, like the one we just solved, are often used to model such phenomena. By understanding how to solve these equations, you can predict future population sizes or the amount of remaining radioactive material after a certain period. This skill is crucial in fields like biology, environmental science, and nuclear physics.
