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3 years ago

1 ln(1) + 2 ln(2) + 3 ln(3) + ⋯ + 10 ln(10)

Mathematics

1 answer:

Lapatulllka [165]3 years ago

6 0

Answer:

$\ln(1\cdot2^{2} \cdot3^{3}\cdot4^{4} ...\cdot10^{10} )$

Step-by-step explanation:

By logarithm rules

$\ln1+2\ln2+3\ln3+....+10\ln10\\\ln1+\ln2^2+\ln3^3+....\ln10^{10}\\\ln(1\cdot2^{2} \cdot3^{3}\cdot4^{4} ...\cdot10^{10} )$

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4/16 in simplest form

vova2212 [387]

4 goes into 4 once, and then into 16 4 times. Making the final answer 1/4

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3 years ago

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What is the true solution to the equation below?

Marina CMI [18]

Answer:

The solution is:

$x=4$

Step-by-step explanation:

Considering the expression

$lne^{lnx}+lne^{lnx}^{^2}=2ln8$

$\ln \left(e^{\ln \left(x\right)}\right)+\ln \left(e^{\ln \left(x\right)\cdot \:2}\right)=2\ln \left(8\right)$

$\mathrm{Apply\:log\:rule}:\quad \:log_a\left(a^b\right)=b$

$\ln \left(e^{\ln \left(x\right)}\right)=\ln \left(x\right),\:\space\ln \left(e^{\ln \left(x\right)2}\right)=\ln \left(x\right)2$

$\ln \left(x\right)+\ln \left(x\right)\cdot \:2=2\ln \left(8\right)$

$\mathrm{Add\:similar\:elements:}\:\ln \left(x\right)+2\ln \left(x\right)=3\ln \left(x\right)$

$3\ln \left(x\right)=2\ln \left(8\right)$

$\mathrm{Divide\:both\:sides\:by\:}3$

$\frac{3\ln \left(x\right)}{3}=\frac{2\ln \left(8\right)}{3}$

$\ln \left(x\right)=\frac{2\ln \left(8\right)}{3}.....A$

Solving the right side of the equation A.

$\frac{2\ln \left(8\right)}{3}$

$\ln \left(8\right):\quad 3\ln \left(2\right)$

Because

$\ln \left(8\right)$

$\mathrm{Rewrite\:}8\mathrm{\:in\:power-base\:form:}\quad 8=2^3$

⇒ $\ln \left(2^3\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

$\ln \left(2^3\right)=3\ln \left(2\right)$

$\frac{2\ln \left(8\right)}{3}=\frac{2\cdot \:3\ln \left(2\right)}{3}$

$\mathrm{Multiply\:the\:numbers:}\:2\cdot \:3=6$

$=\frac{6\ln \left(2\right)}{3}$

$\mathrm{Divide\:the\:numbers:}\:\frac{6}{3}=2$

$=2\ln \left(2\right)$

So, equation A becomes

$\ln \left(x\right)=2\ln \left(2\right)$

$\mathrm{Apply\:log\:rule}:\quad \:a\log _c\left(b\right)=\log _c\left(b^a\right)$

$=\ln \left(2^2\right)$

$=\ln \left(4\right)$

$\ln \left(x\right)=\ln \left(4\right)$

$\mathrm{Apply\:log\:rule:\:\:If}\:\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\:\mathrm{then}\:f\left(x\right)=g\left(x\right)$

$x=4$

Therefore, the solution is

$x=4$

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3 years ago

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3 years ago

At a toy factory, toy cars are produced on an assembly line. It takes 2.054 s to attach each of the four wheels. It takes 3.652

IrinaK [193]

Add together the time it takes to put on the four wheels and the hood.
2.054s +3.652s = 5.704s

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5.706s times 42= 239.652s

answer number two is correct

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3 years ago

Find the value of x if 212x5 is a multiple of 3 and 11 please find out now guys HELP!

sammy [17]

Answer:

x = 8

Step-by-step explanation:

If the number 212x5 is a multiple of 3, the sum of the digits needs to be a multiple of 3, so we have:

2 + 1 + 2 + x + 5 = 10 + x

The value of x can be 2, 5 or 8

If the number is a multiple of 11, we have to calculate the first number minus the second, plus the third, minus the fourth... and so on, and the result needs to be a multiple of 11. So we have:

2 - 1 + 2 - x + 5 = 8 - x

x needs to be 8, so we have 8 - 8 = 0, that is a multiple of 11.

So the value of x is 8.

8 0

3 years ago