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3^x=x^5 what numbers make this true please give answers to the 2nd decimal

Guest May 2, 2015

Best Answer 

 #1
avatar+86919 
+10

3^x  = x ^5       take the log of both sides

log 3^x   = logx^5     and we can write

x log3  = 5 log x

Rearrange

x / log x  = 5/log 3

 

Mmmm....that doesn't look too promising, does it???

Let's just graph the original functions and see where they meet (if they do....!!)

Here's a partial graph......https://www.desmos.com/calculator/roelalpkth

The graph shows that one solution occurs at about (1.343, 4.375)

Another solution occurs at about (10.351, 150,462).....shown here....https://www.desmos.com/calculator/ujgxybuyeq

 

This happens because from x values (-∞, 1.343), the graph of 3^x > x^5. Then  from (1.343, 10.351), the graph of x^5 > 3^x.  Finally, the graph of 3^x "overtakes" the graph of x^5 at about x = 10.351, and is always greater at any comparavle x value after that - as we would expect....!!!!

 

 

  

CPhill  May 2, 2015
 #1
avatar+86919 
+10
Best Answer

3^x  = x ^5       take the log of both sides

log 3^x   = logx^5     and we can write

x log3  = 5 log x

Rearrange

x / log x  = 5/log 3

 

Mmmm....that doesn't look too promising, does it???

Let's just graph the original functions and see where they meet (if they do....!!)

Here's a partial graph......https://www.desmos.com/calculator/roelalpkth

The graph shows that one solution occurs at about (1.343, 4.375)

Another solution occurs at about (10.351, 150,462).....shown here....https://www.desmos.com/calculator/ujgxybuyeq

 

This happens because from x values (-∞, 1.343), the graph of 3^x > x^5. Then  from (1.343, 10.351), the graph of x^5 > 3^x.  Finally, the graph of 3^x "overtakes" the graph of x^5 at about x = 10.351, and is always greater at any comparavle x value after that - as we would expect....!!!!

 

 

  

CPhill  May 2, 2015
 #2
avatar+92625 
0

I do not know how to do this any other way either Chris :)

Melody  May 3, 2015
 #3
avatar+26722 
+5

Here's a numerical approach using Newton-Raphson:

 3^x = x^5

(These basically agree with Chris's solutions, though I think he misprinted 10.851 as 10.351.)

 

To two decimal places these are  1.34 and 10.85

 

.

Alan  May 3, 2015
 #4
avatar+86919 
0

LOL!!!.....I didn't misprint anything......I just can't see........!!!!

10.851 is correct......

 

  

CPhill  May 3, 2015

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