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3 years ago

15. Explain why we need to specify 0 < b < 1 and b > 1 as valid values for the base b in the expression logb(xx).

Mathematics

1 answer:

damaskus [11]3 years ago

3 0

Answer:

The base (b) has to be positive and different of 1. The logarithm is the inverse of exponential, so:

logb(a) = x ⇒ a = bˣ

So, for b = 0 ⇒ 0ˣ = a

And there is impossible, "a" only could be 0.

For b = 1 ⇒ 1ˣ = a

And the same thing would happen, the logarithming would be to be 1, and the function will be extremally restricted.

For b<0, then the expression a = bˣ will be also restricted, and will not represent all values of a.

So, 0<b<1 and b >1.

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3 years ago

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A low-strength children’s/adult chewable aspirin tablet contains 81 mg of aspirin per tablet. How many tablets may be prepared f

Svetlanka [38]

Answer:

12,345 tablets may be prepared from 1 kg of aspirin.

Step-by-step explanation:

The problem states that low-strength children’s/adult chewable aspirin tablets contains 81 mg of aspirin per tablet. And asks how many tablets may be prepared from 1 kg of aspirin.

Since the problem measures the weight of a tablet in kg, the first step is the conversion of 81mg to kg.

Each kg has 1,000,000mg. So

1kg - 1,000,000mg

xkg - 81mg.

1,000,000x = 81

$x = \frac{81}{1,000,000}$

x = 0.000081kg

Each tablet generally contains 0.000081kg of aspirin. How many such tablets may be prepared from 1 kg of aspirin?

1 tablet - 0.000081kg

x tablets - 1kg

0.000081x = 1

$x = \frac{1}{0.000081}$

x = 12,345 tablets

12,345 tablets may be prepared from 1 kg of aspirin.

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3 years ago

If the length of a rectangle is 3 and the width is 2, the length of the diagonal is a 13 b 5 c √5 d √13

trasher [3.6K]

Answer:

The length of the rectangle is 12cm and the area of the rectangle is 60cm2.

Explanation:

By definition, the angles of a rectangle are right. Therefore, drawing a diagonal creates two congruent right triangles. The diagonal of the rectangle is the hypotenuse of the right triangle. The sides of the rectangle are the legs of the right triangle. We can use the Pythagorean Theorem to find the unknown side of the right triangle, which is also the unknown length of the rectangle.

Recall that the Pythagorean Theorem states that the sun of the squares of the legs of a right triangle is equal to the square of the hypotenuse. a2+b2=c2

52+b2=132

25+b2=169

25−25+b2=169−25

b2=144

√b2=√144

b=±12

Since the length of the side is a measured distance, the negative root is not a reasonable result. So the length of the rectangle is 12 cm.

The area of a rectangle is given by multiplying the width by the length.

A=(5cm)(12cm)

A=60cm2

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3 years ago

Joe is building a post for his mailbox. To find the correct dimensions, he needs to expand this expression: (x-3) (x-9) (x-4) Se

kolbaska11 [484]

Answer:

$(x-3)(x-9)(x-4)=x^3-16x^2+75x-108$

Step-by-step explanation:

Given:

The expression to expand is given as:

$(x-3)(x-9)(x-4)$

Let us expand the first two binomials of the given expression using FOIL method.

The FOIL method states that:

$(a + b)(c + d) = ac + ad + bc + bd$

$(x-3)(x-9)=(x\times x)+(x\times -9)+(-3\times x)+(-3\times -9)\\\\(x-3)(x-9)=x^2-9x-3x+27=x^2-12x+27$

Now, let us multiply the result with the remaining binomial. Multiplying each term of the trinomial with each term of the binomial, we get:

$(x^2-12x+27)(x-4)\\\\= (x^2\times x)+ (x^2\times -4)+(-12x\times x)+(-12x\times -4)+(27\times x)+(27\times -4)\\\\=x^3-4x^2-12x^2+48x+27x-108\\\\=x^3-16x^2+75x-108...........(\because-12x^2-4x^2=-16x^2,48x+27x=75x)$

Therefore, the equivalent expression after expanding is given as:

$(x-3)(x-9)(x-4)=x^3-16x^2+75x-108$

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3 years ago