- #1
DrLiangMath
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Solve $4^x+3^x=12$ without using CAS?
Is There an Exact Solution to the Equation \(4^x + 3^x = 12\)?
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In summary, the person asked Dan to solve the equation, but Dan said he couldn't because he can't solve it with a CAS.
- #2
maxkor
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- #3
DrLiangMath
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Nice solution! But if you could find an exact solution it would be much better.maxkor said:
- #4
maxkor
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Do you have an answer?MathTutoringByDrLiang said:
- #5
DrLiangMath
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maxkor said:
I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.
- #6
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I can solve something like $4^x + 6^x = 9^x$ only because I can put it into a homogeneous format. I can't think of any way to do this with $3^x + 4^x = 12$. Even if there were a typo I don't think we can do $3^x + 4^x = 12^x$.MathTutoringByDrLiang said:
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- #7
Office_Shredder
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If you plug it into wolfram alpha, I don't think you can solve this with a CAS either, it's just numeric approximations for you.
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DrLiangMath said:
Reminds me of...
Fermat said:
FAQ: Is There an Exact Solution to the Equation \(4^x + 3^x = 12\)?
How do I solve an exponential equation?
Solving an exponential equation involves using logarithms to isolate the variable. First, take the logarithm of both sides of the equation. Then, use algebraic manipulation to solve for the variable. Finally, plug the solution back into the original equation to check for accuracy.
What is the difference between a linear and an exponential equation?
A linear equation has a constant rate of change, while an exponential equation has a variable rate of change. In a linear equation, the variable is raised to the first power, while in an exponential equation, the variable is raised to a power greater than 1.
Can I use a calculator to solve an exponential equation?
Yes, you can use a calculator to solve an exponential equation. Most scientific calculators have a "log" or "ln" function that can be used to take the logarithm of a number. However, it is important to understand the steps involved in solving an exponential equation by hand in order to use the calculator correctly.
Are there any special rules for solving exponential equations?
Yes, there are a few special rules for solving exponential equations. One is the power rule, which states that when raising a power to another power, you multiply the exponents. Another rule is the product rule, which states that when multiplying two exponential expressions with the same base, you can add the exponents. Additionally, there is a rule for solving exponential equations with the same base on both sides, which involves setting the exponents equal to each other.
What are some real-life applications of exponential equations?
Exponential equations are used in many fields, including finance, biology, and physics. One common application is compound interest, where the amount of money in an account grows exponentially over time. In biology, exponential growth is used to model population growth. In physics, exponential decay is used to model radioactive decay. Additionally, exponential equations are used in computer science and technology to model data growth and processing speed.