Just a simple equation!!!

Hello, i would like to provide me a full solution for this equation 2^x + 3^x = 2 I know that the answer is x = 0 but i would like to tell me how i solve it step by step I wonder also if this is a right way to solve it 2^X + 3^X = 1 + 1 => 2^X + 3^X = 2^0 + 3^0 => 2^X = 2^0 AND 3^x = 3^0 so x = 0

Note by Kon Tim
8 years, 2 months ago

No vote yet
7 votes

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

ddx(2x+3x)=ln(2)2x+ln(3)3x\frac{d}{dx}(2^x+3^x)=ln(2)2^x+ln(3)3^x .

ln(2)2x+ln(3)3x>0ln(2)2^x+ln(3)3^x > 0 so the gradient of 2x+3x2^x+3^x is always positive, so 2x+3x2^x+3^x must be increasing. Hence 2x+3x2^x+3^x can only cross the line y=2y=2 once, so 2x+3x=22^x+3^x=2 can only have one solution. By observation, x=0x=0 is a solution, and so must be the only solution.

Clifford Wilmot - 8 years, 2 months ago

Log in to reply

How you guys latexify the problem at brilliant??

Advitiya Brijesh - 8 years, 2 months ago

Log in to reply

brilliant.org/math-formatting-guide/

Clifford Wilmot - 8 years, 2 months ago

Log in to reply

@Clifford Wilmot – $\text{Thanks}$

Advitiya Brijesh - 8 years, 2 months ago

Log in to reply

@Advitiya Brijesh – \text {Thanks}

Advitiya Brijesh - 8 years, 2 months ago

Log in to reply

@Advitiya Brijesh – ThanksThanks

Use \ (Thanks) without the first space.

Zi Song Yeoh - 8 years, 2 months ago

Log in to reply

@Zi Song Yeoh – It can also be \ (\text{Thanks}\ ) without spaces which would appear as Thanks\text{Thanks} :)

Nishanth Hegde - 8 years, 2 months ago

Log in to reply

@Nishanth Hegde – $\ (\text{Thanks}\ )$

Advitiya Brijesh - 8 years, 2 months ago

Log in to reply

@Advitiya Brijesh – $text\(THANKS)$

superman son - 8 years, 1 month ago

@Zi Song Yeoh – (Thanks)

superman son - 8 years, 1 month ago

@Clifford Wilmot – no latex..

Advitiya Brijesh - 8 years, 2 months ago

this is just my own step but you can ask if there is something you want to add :)

2x+3x=22^{x} + 3^{x} = 2

Transposing 2x2^{x} to the right side

3x=2−2x3^{x} = 2-2^{x}

by the use of logarithm

log3(2−2x)=xlog_{3}(2-2^{x}) = x

by using the law of logarithm, (2−2x)(2 - 2^{x}) must be greater than zero because you cannot log the number 0 or any negative number, meaning x<1x<1

so the only integer that satisfies the condition is 0 so x=0x=0

Ian Mana - 8 years, 2 months ago

Log in to reply

Thank you for your reply but i cant take for granted that the solution is an integer.

Kon Tim - 8 years, 2 months ago

Log of negative numbers is defined, as Calvin Sir has mentioned in some posts. Only log of 0 is not defined.

Sambit Senapati - 8 years, 2 months ago
×

Problem Loading...

Note Loading...

Set Loading...