Help me with this math problem?

How to solve this equation?

2^(x)+3^(x)=13

I know that the answer is x=2 but what are the ways of solving this type of problem?

4 Answers

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  • Alan
    Lv 7
    11 months ago
    Favourite answer

    One way

    1st graph: y = 2^x + 3^x -13   

    Look for zero crossing  

    Use mean value theorem 

    then you know that it is between two values 

    for examples 

    y for (x=1) = 2^1 + 3^1 -13 =   5-13= -8 

    y for (x=3)  = 2^3 + 3^3 -13 =  8+27-13 = 22  

    so that tells you it is between 1 and 3 

    try 1.5 and 2.5 

    y(1.5) =   2^(3/2) + 3^(3/2) -13 =  -4.9754204525

    y(2.5) = 2^(2.5)+ 3^(2.5) -13 =  8.2453115176

    so it is between 1.5 and 2.5  

    Now use Newton-Raphson Method to  

    narrow the answer even more 

    try 2.1 as the answer 

    f(x)= 2^x + 3^x -13 

    f'(x) = ln(2)*2^x  + ln(3)*3^x  -13  

    x_next= x_prev -   f(x)/f'(x) =

    x_next = x_prev -    (2^x + 3^x -13)/ (  ln(2)*2^x + ln(3)*3^x -13  )

    If you use the formula

    2.1  next guess  2.0048920505

    2.0048920505  next guess 

    2.0000120647 next guess  2.0000000001

    2.0000000001  next guess     2

    Attachment image
  • sepia
    Lv 7
    11 months ago

    2^(x) + 3^(x) = 13

    x = 2 is the answer.

    2^3 + 3^3 =35

    2^2 + 3^2 = 13

    2^1 + 3^1 = 5

  • 11 months ago

    2^(x) + 3^(x) = 13

    By trial and error:

    x = 2

  • Anonymous
    11 months ago

    There is no way to solve algebraically. 

    If it is know the answer is an integer You could start at 0 and work your way up.

     

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