What is the slope of the line that passes through the points listed in the table?

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

A serving of chickpeas contains 2,150 milligrams of potassium. How many grams of

Zepler [3.9K]

Answer: 109 mg

Step-by-step explanation:

5 0

2 years ago

Raul uses a credit card to take out a 3-year loan of $4500 at a rate of 15% to purchase

mars1129 [50]

Answer:

Amount of loan pay = $6.840 (Approx.)

Step-by-step explanation:

Given:

Amount of loan borrow = $4,500

Rate of interest = 15% = 0.15

Number of year = 3 year

Find:

Amount of loan pay

Computation:

Future value = Present value[1+r]ⁿ

Amount of loan pay = Amount of loan borrow[1+r]ⁿ

Amount of loan pay = 4,500[1+0.15]³

Amount of loan pay = 4,500[1.15]³

Amount of loan pay = 4,500[1.52]

Amount of loan pay = $6.840 (Approx.)

7 0

2 years ago

I really need help! can someone help me

Jet001 [13]

Answer:

M + $13.72 = $58.22

Step-by-step explanation:

Manny and Sam combined (+) have a total (=) of $58.22

M + S = $58.22

M + $13.72 = $58.22

8 0

2 years ago

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HELP THIS IS DUE SOON!<br><br> find the error!

Rudik [331]

Answer:

Step 1

Step-by-step explanation:

they were supposed to multiply 2x by 4 and -1 by 4 but they only multiplied 2x by 4.

3 0

2 years ago

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