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3 years ago

What is the area of 2, 1/3, 4/3 and 2/9.

Mathematics

2 answers:

OLga [1]3 years ago

8 0

Answer: 35/3 or 11 2/3

Step-by-step explanation:

NISA [10]3 years ago

8 0

The answer is 35/3 Or 11 2/3

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If a,b,c and d are positive real numbers such that logab=8/9, logbc=-3/4, logcd=2, find the value of logd(abc)

Eva8 [605]

We can expand the logarithm of a product as a sum of logarithms:

$\log_dabc=\log_da+\log_db+\log_dc$

Then using the change of base formula, we can derive the relationship

$\log_xy=\dfrac{\ln y}{\ln x}=\dfrac1{\frac{\ln x}{\ln y}}=\dfrac1{\log_yx}$

This immediately tells us that

$\log_dc=\dfrac1{\log_cd}=\dfrac12$

Notice that none of $a,b,c,d$ can be equal to 1. This is because

$\log_1x=y\implies1^{\log_1x}=1^y\implies x=1$

for any choice of $y$ . This means we can safely do the following without worrying about division by 0.

$\log_db=\dfrac{\ln b}{\ln d}=\dfrac{\frac{\ln b}{\ln c}}{\frac{\ln d}{\ln c}}=\dfrac{\log_cb}{\log_cd}=\dfrac1{\log_bc\log_cd}$

so that

$\log_db=\dfrac1{-\frac34\cdot2}=-\dfrac23$

Similarly,

$\log_da=\dfrac{\ln a}{\ln d}=\dfrac{\frac{\ln a}{\ln b}}{\frac{\ln d}{\ln b}}=\dfrac{\log_ba}{\log_bd}=\dfrac{\log_db}{\log_ab}$

so that

$\log_da=\dfrac{-\frac23}{\frac89}=-\dfrac34$

So we end up with

$\log_dabc=-\dfrac34-\dfrac23+\dfrac12=-\dfrac{11}{12}$

###

Another way to do this:

$\log_ab=\dfrac89\implies a^{8/9}=b\implies a=b^{9/8}$

$\log_bc=-\dfrac34\implies b^{-3/4}=c\implies b=c^{-4/3}$

$\log_cd=2\implies c^2=d\implies\log_dc^2=1\implies\log_dc=\dfrac12$

Then

$abc=(c^{-4/3})^{9/8}c^{-4/3}c=c^{-11/6}$

So we have

$\log_dabc=\log_dc^{-11/6}=-\dfrac{11}6\log_dc=-\dfrac{11}6\cdot\dfrac12=-\dfrac{11}{12}$

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3 years ago

What is 3.25% written as a fraction in lowest terms?

pantera1 [17]

13/400 is the answer

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3 years ago

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Does the point(1, 8)satisfy the inequality y ≥ 6x + 2 ?

nalin [4]

Answer: Yes

Step-by-step explanation:

1. Substitute 1 for x

2. 6 times 1 is 6 plus 2 is 8

3.8 is equal to or greater than 8

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3 years ago

Factor these expressions into an equivalent form.

Marrrta [24]

(c+8)(c-8)

3(2y+5)(2y-5)

There are no like terms so it's still ab^2-b

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Nat2105 [25]

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3 years ago

What is the area of 2, 1/3, 4/3 and 2/9.​

What is the area of 2, 1/3, 4/3 and 2/9.