Two thirds of a number decreased by 27 is at least 9

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

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

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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

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

4 0

2 years ago

Is 55 miles per hour a unit rate ?

sveticcg [70]

Answer: wait I was wrong I saw thinking about something else

yes that is a unit rate.

3 0

2 years ago

Read 2 more answers

An insect crawls 30 inches in 16.3 minutes. If

omeli [17]

Answer:

Step-by-step explanation:

Convert 6 hours to minutes so that 6 hours and 16. 3 minutes can be in the same units.

Since 1 hour = 60 minutes, then 6 hours = 6 x 60 minutes = 360 minutes

Since the insect has more minutes to crawl (360 minutes), then we expect more inches to be covered. Therefore, the ratio is 360 minutes to 16.3 minutes

= 360 minutes : 16.3 minutes

= 360 : 16.3

Now express the ratio as fraction and relate it with 30 inches

= 360/16.3 x 30/1

= 10800/16.3

= 662 .576687 and this is approximately 650 inches

Answer = 650 = (C)

3 0

2 years ago

Need help it’s easy

defon

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2 years ago

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Mr. Rifkin had 240 digital and manual cameras. After donating 82 digital cameras and 26 manual cameras, he had 3 times as many d

Oliga [24]

<span>He started with 184 digital cameras and 56 manual cameras

Work--> </span>d + m = 240 equation 1 (d-82) = 3(m-26) equation 2d - 82 = 3m - 78d - 3m = 82 - 78d - 3m = 16 equation 2 simplified d + m = 240d - 3m = 16 subtract equation 1 from equation 2--------------- -4m = -224 m = -224/-4 m = 56 d + m = 240d = 240-md = 240-56d = 184

4 0

3 years ago