Type the number 3.14×103 in standard notation. express your answer using standard notation.

If <img src="https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20x%20%3D%20log_%7Ba%7D%28bc%29" id="TexFormula1" title="\rm \: x = log_{a}(

timama [110]

Use the change-of-basis identity,

$\log_x(y) = \dfrac{\ln(y)}{\ln(x)}$

to write

$xyz = \log_a(bc) \log_b(ac) \log_c(ab) = \dfrac{\ln(bc) \ln(ac) \ln(ab)}{\ln(a) \ln(b) \ln(c)}$

Use the product-to-sum identity,

$\log_x(yz) = \log_x(y) + \log_x(z)$

to write

$xyz = \dfrac{(\ln(b) + \ln(c)) (\ln(a) + \ln(c)) (\ln(a) + \ln(b))}{\ln(a) \ln(b) \ln(c)}$

Redistribute the factors on the left side as

$xyz = \dfrac{\ln(b) + \ln(c)}{\ln(b)} \times \dfrac{\ln(a) + \ln(c)}{\ln(c)} \times \dfrac{\ln(a) + \ln(b)}{\ln(a)}$

and simplify to

$xyz = \left(1 + \dfrac{\ln(c)}{\ln(b)}\right) \left(1 + \dfrac{\ln(a)}{\ln(c)}\right) \left(1 + \dfrac{\ln(b)}{\ln(a)}\right)$

Now expand the right side:

$xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} \\\\ ~~~~~~~~~~~~+ \dfrac{\ln(c)\ln(a)}{\ln(b)\ln(c)} + \dfrac{\ln(c)\ln(b)}{\ln(b)\ln(a)} + \dfrac{\ln(a)\ln(b)}{\ln(c)\ln(a)} \\\\ ~~~~~~~~~~~~ + \dfrac{\ln(c)\ln(a)\ln(b)}{\ln(b)\ln(c)\ln(a)}$

Simplify and rewrite using the logarithm properties mentioned earlier.

$xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} + \dfrac{\ln(a)}{\ln(b)} + \dfrac{\ln(c)}{\ln(a)} + \dfrac{\ln(b)}{\ln(c)} + 1$

$xyz = 2 + \dfrac{\ln(c)+\ln(a)}{\ln(b)} + \dfrac{\ln(a)+\ln(b)}{\ln(c)} + \dfrac{\ln(b)+\ln(c)}{\ln(a)}$

$xyz = 2 + \dfrac{\ln(ac)}{\ln(b)} + \dfrac{\ln(ab)}{\ln(c)} + \dfrac{\ln(bc)}{\ln(a)}$

$xyz = 2 + \log_b(ac) + \log_c(ab) + \log_a(bc)$

$\implies \boxed{xyz = x + y + z + 2}$

(C)

6 0

2 years ago

Draw triangle abc where:

Mariulka [41]

<u>Step-by-step explanation:</u>

We are asked to draw a triangle.

The information we are given with is AB is a line segment with length 3.9 cm.

∠CAB = 50°.

∠ABC = 63°.

With the given details we need to draw triangle using given angles. We need protractor and ruler to draw the triangle.

Step 1: Draw the line segment of length 3.9 cm using ruler and mark as AB.

Step 2: Using protractor, we need to mark the angle. Keep the center of the protractor at the A and mark with a point at 50°.

Step 3: Draw a line from a passing through the marked point.

Step 4: Repeat step 2 with marking at 63° and step 3 on the other side.

Step 5: Mark the intersection point as C.

8 0

3 years ago

How many classes have between 27 and 31 students?

bekas [8.4K]

I’m not sure but I think 4 classes have between 27 and 31 students

5 0

3 years ago

a farmer plant white rice and brown rice on 10 acres and he has 18 liter of pesticide to use. white rice requires 2 liters of pe

exis [7]

Answer:

Step-by-step explanation:

sticide per acre and brown rice requires 1 liter of pesticide per acre. if he can earn $5000 for each acre of white rice ans $3000 for each acre of brown rice, how many acre of each should by plan to maximize his earnings? what are his maximum earning?

6 0

3 years ago

[PICTURE ATTACHED] HELP PLEASE

aleksandr82 [10.1K]

Answer:

I can barely see

Step-by-step explanation:

6 0

3 years ago

Read 2 more answers