SOLUTION: The relation between the number of time in days, D, it takes for a planet to revolve around the Sun is related to the planet’s average distance from the Sun, k, in millions of ki

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Question 1205558: The relation between the number of time in days, D, it takes for a planet to revolve around the Sun is related to the planet’s average distance from the Sun, k, in millions of kilometres is defined by logD = 3/2 logk-0.7. How much longer does it take Mars than Venus to orbit the Sun if Mars is 207 million km and Venus is 108 million km away from the Sun?
equation orientation: https://gyazo.com/61b74df513162501c4701a08f0cbab60
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Mars is k=207 million km and
Venus is k=108 million km away from the Sun
Mars:
log%28D%29=%283%2F2%29log%28207%29-0.7
log%28D%29=log%28207%5E%283%2F2%29%29-log%2810%5E0.7%29
log%28D%29=log%28207%5E%283%2F2%29%2F%2810%5E0.7%29%29
log%28D%29=log%28594.231%29
D=594.231 days

Venus:
log%28D%29=%283%2F2%29log%28108%29-0.7
log%28D%29=log%28108%5E%283%2F2%29%29-log%2810%5E0.7%29
log%28D%29=log%28108%5E%283%2F2%29%2F%2810%5E0.7%29%29
log%28D%29=log%28223.942%29
D=223.942 days

594.231-223.942=370.289+
it takes Mars 370.289 days longer than Venus to orbit the Sun


Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
The relation between the number of time in days, D, it takes for a planet to revolve around the Sun
is related to the planet’s average distance from the Sun, k, in millions of kilometres is defined by
logD = 3/2 logk-0.7. How much longer does it take Mars than Venus to orbit the Sun
if Mars is 207 million km and Venus is 108 million km away from the Sun?
equation orientation: https://gyazo.com/61b74df513162501c4701a08f0cbab60
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            See how simple the calculations are. 


Time for Mars to make one revolution around the Sun

      log%28%28D%5BMars%5D%29%29 = %283%2F2%29%2Alog%28%28207%29%29 - 0.7 = use your calculator = 

    = %283%2F2%29%2A%282.315970345%29 - 0.7 = use your calculator, again = 2.773955518.


It gives for Mars

      D%5BMars%5D = 10%5E2.773955518 = use your calculator = 594 days (rounded).



             Similar calculations for Venus



Time for Venus to make one revolution around the Sun

      log%28%28D%5BVenus%5D%29%29 = %283%2F2%29%2Alog%28%28108%29%29 - 0.7 = use your calculator = 

    = %283%2F2%29%2A%282.033423755%29 - 0.7 = use your calculator, again = 2.350135633
.


It gives for Venus

      D%5BVenus%5D = 10%5E2.350135633 = use your calculator = 224 days (rounded).


The problem asks about the difference 

      D%5BMars%5D - D%5BVenus%5D = 594 - 224 = 370 days  (approximately).    ANSWER

Solved.

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The major goal of this problem is to teach you using your calculator for similar tasks.

First,  you make straightforward calculations to find the logarithm values,  using given formula.

Next,  you potentiate to restore the numbers from the found logarithm values.


In this problem,  there is  NO  ANY  NEED  to manipulate with formulas for logarithms.

This problem is not intended for such manipulations.