To solve the given equation, w need to review some rules:
![(1) \ ln \ e = 1 \\ (2) \ ln \ b^{a} = a*ln \ b \\ (3) \ ln \ a + ln \ b = ln \ (ab) \\ (4) \ ln \ a - ln \ b = ln \ \frac{a}{b} \\](https://tex.z-dn.net/?f=%281%29%20%5C%20ln%20%5C%20e%20%3D%201%20%5C%5C%20%0A%282%29%20%5C%20ln%20%5C%20%20b%5E%7Ba%7D%20%3D%20a%2Aln%20%5C%20b%20%5C%5C%0A%283%29%20%5C%20ln%20%5C%20a%20%2B%20ln%20%5C%20b%20%3D%20ln%20%5C%20%28ab%29%20%5C%5C%0A%284%29%20%5C%20ln%20%5C%20a%20-%20ln%20%5C%20b%20%3D%20ln%20%5C%20%20%5Cfrac%7Ba%7D%7Bb%7D%20%20%5C%5C)
The given equation is :
![ln \ e^{ln \ x} + ln \ e^{ln \ x^{2}} = 2 \ ln \ 8](https://tex.z-dn.net/?f=ln%20%5C%20%20e%5E%7Bln%20%5C%20x%7D%20%2B%20ln%20%5C%20%20e%5E%7Bln%20%5C%20%20x%5E%7B2%7D%7D%20%3D%202%20%5C%20ln%20%5C%208)
![ln \ x * ln \ e + ln \ x^{2} * ln \ e = 2 \ ln \ 8](https://tex.z-dn.net/?f=ln%20%5C%20x%20%2A%20ln%20%5C%20e%20%2B%20ln%20%5C%20%20x%5E%7B2%7D%20%2A%20ln%20%5C%20e%20%3D%202%20%5C%20ln%20%5C%208%20)
⇒⇒⇒⇒ rule (2)
![ln \ x + ln \ x^{2} = ln \ 8^{2}](https://tex.z-dn.net/?f=ln%20%5C%20x%20%2B%20ln%20%5C%20%20x%5E%7B2%7D%20%3D%20ln%20%5C%20%208%5E%7B2%7D%20)
⇒⇒⇒⇒ rule (1) and rule (2)
![ln \ ( x* x^{2} ) = ln \ 64](https://tex.z-dn.net/?f=ln%20%5C%20%28%20x%2A%20%20x%5E%7B2%7D%20%29%20%3D%20ln%20%5C%2064)
⇒⇒⇒⇒ rule (3)
removing the nature logarithm from both sides
![x^{3} = 64 = 4^{3}](https://tex.z-dn.net/?f=%20x%5E%7B3%7D%20%3D%2064%20%3D%20%204%5E%7B3%7D%20%20)
∴ x = 4
So, the correct answer is option (2) ⇒⇒⇒⇒ x = 4
Answer:
3
1
2
Step-by-step explanation:
Math simplification
Answer:
60
Step-by-step explanation:
After adding 88 red marbles, there are 50 blue marbles out of the 204 marbles total.
Let the number of blue marbles to be added be x.
![\frac{50+x}{204+x}=\frac{5}{12} \\ \\ 600+12x=1020+5x \\ \\ 420=7x \\ \\ x=60](https://tex.z-dn.net/?f=%5Cfrac%7B50%2Bx%7D%7B204%2Bx%7D%3D%5Cfrac%7B5%7D%7B12%7D%20%5C%5C%20%5C%5C%20600%2B12x%3D1020%2B5x%20%5C%5C%20%5C%5C%20420%3D7x%20%5C%5C%20%5C%5C%20x%3D60)
Answer:
look at the picture i have sent
The answer is 9
Explanation:
We know the total amount of spaghetti sauce is 20.5 cups; additionally, each container can be filled with a maximum of 2.5 cups of this sauce. In this context, the process to know the number of containers Mr. Rush needs to store all the sauce is to divide the total amount by the amount each container can hold.
20.5 cups of sauce ÷ 2.5 cups per container = 8.2 containers
This means 8 containers can be completely filled and a ninth container is required to hold the small amount of the sauce missing (0.2)