- #1
Champdx
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Homework Statement
2^x+3^x=13, x=2. How do i prove it?
The Attempt at a Solution
i did this
x log 2 + x log 3 = log 13
x(log 2 + log 3)=log 13
x(0.301+0.477)= 1.11
0.778x=1.11
x=1.43
Champdx said:x log 2 + x log 3 = log 13
You can't get the equation above from the one you started with. log(A + B) [itex]\neq[/itex] logA + logB. What you did was to take the log of both sides to getChampdx said:Homework Statement
2^x+3^x=13, x=2. How do i prove it?
The Attempt at a Solution
i did this
x log 2 + x log 3 = log 13
Champdx said:x(log 2 + log 3)=log 13
x(0.301+0.477)= 1.11
0.778x=1.11
x=1.43
tiny-tim said:If the question says "solve 2x + 3x = 13", then there's no exact way of doing it, you'll have to use an approximation method (or guess).
If the question says "show that 2x + 3x = 13 has a solution x = 2", or "prove that 2x + 3x = 13 has a solution x = 2", then all you need to do is to show that 22 + 32 = 13.
To begin, we can substitute x=2 into the equation, giving us 2^2+3^2=13. This simplifies to 4+9=13, which is a true statement. This is our starting point for the proof.
Yes, we can use algebra to prove this equation. We can start by rewriting 2^x+3^x as (2+3)^x. Then, using the binomial theorem, we can expand this to (2^x+3^x)(2+3)^(x-1). From here, we can simplify the equation and show that it is equal to 13 for x=2.
In addition to algebra, we can also use logarithms to prove this equation. We can start by taking the logarithm of both sides, giving us log(2^x+3^x)=log(13). Then, using logarithm properties, we can rewrite the left side as xlog2+xlog3=log(13). From here, we can solve for x and show that it is equal to 2.
Yes, we can also use a graph to prove this equation. We can graph the functions y=2^x+3^x and y=13 and show that they intersect at x=2. This visually demonstrates that the two expressions are equal for x=2.
Proving this equation is significant because it shows that the two expressions, 2^x+3^x and 13, are equivalent for the specific value of x=2. This can also lead to a deeper understanding of exponential and logarithmic functions and their properties.