# 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

- AlanLv 711 months agoFavourite 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

- sepiaLv 711 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

- Anonymous11 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.