# Simple Indices

## 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

## Answers and Replies

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HallsofIvy
Homework Helper
??? Why not just substitute "2" for "x"?

$2^x+ 3^x= 2^2+ 3^2= 4+ 9= 13$.

You are not asked so solve the equation!

tiny-tim
Homework Helper
Hi Champdx!

(try using the X2 tag just above the Reply box )
x log 2 + x log 3 = log 13
Nooo

that's log (x2x3)

Actually i know that the answer x=2 but how am i suppose to prove it?

tiny-tim
Homework Helper
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.

Mark44
Mentor

## 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
You can't get the equation above from the one you started with. log(A + B) $\neq$ logA + logB. What you did was to take the log of both sides to get
log(2^x + 3^x) = log 13. That's a legitimate step, but it doesn't lead you anywhere.
The problem is that log(2^x + 3^x) $\neq$ log 2^x + log 3^x.
x(log 2 + log 3)=log 13
x(0.301+0.477)= 1.11
0.778x=1.11
x=1.43

Is there any way to solve this question by calculation?

Thanks.

tiny-tim
Homework Helper
Nope.

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.
Champ, read this quote again. There are only two options:
1) "Solve..." or
2) "Show that"/"Prove..."

There's a phrase that comes up frequently in math (that honestly kinda p*sses me off!)...
"By inspection...".
This means "I can SEE the answer, but I can't/won't show HOW you could get the same answer without guessing". There might be a better description out there for this phrase, which I think applies to our situation.

We can see that x = 2 is a solution. We can see that it is a UNIQUE solution because increasing/decreasing x will increase/decrease BOTH terms on the left hand side of the equation...