How to Solve Equations with Unknown Exponents?

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In summary, the conversation is about solving the equation 4^(x+2) = 7^x +681 for x. The speaker is unsure of the procedure and believes logs or natural logs may be involved. They know the answer is 3 from trial and error, but cannot prove it. The expert summarizes that there is no easy way to solve this type of equation and mentions that there is another solution near 4.8622.
  • #1
foxyman20
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Ok i know this should be a relatively simple equation but i haven't done this stuff in so long i can't remember how to solve it. here goes.

4^(x+2) = 7^x +681

Solve for x

I know the answer is 3 from simple trial and error but i can't show the work to prove it and i know that harder problems exist so i would like to know the procedure. I'm pretty sure that either logs or natural logs (LN) is required somehow. Any help would be greatly appreciated.

Thanks.
 
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  • #2
well i figure the physics people should have no problem with this.

Ok i know this should be a relatively simple equation but i haven't done this stuff in so long i can't remember how to solve it. here goes.

4^(x+2) = 7^x +681

Solve for x

I know the answer is 3 from simple trial and error but i can't show the work to prove it and i know that harder problems exist so i would like to know the procedure. I'm pretty sure that either logs or natural logs (LN) is required somehow. Any help would be greatly appreciated.

Thanks.
 
  • #3
there's no easy way to solve an equation that has a polynomial on one side and a transcendental function on the other. Sorry.

P.S. there is another solution near 4.8622 too, but no other solutions besides that one.
 

What is an unknown exponent?

An unknown exponent is a symbol or variable that represents the power to which a base number is raised. It is typically represented by the letter "x" or "n" and is used in exponential equations to find the value of the unknown variable.

How do you solve for unknown exponents?

To solve for unknown exponents, you can use the laws of exponents, which state that when multiplying two numbers with the same base, the exponents are added, and when dividing two numbers with the same base, the exponents are subtracted. You can also use logarithms to solve for an unknown exponent.

What is the difference between a base and an exponent?

The base in an exponential equation is the number that is multiplied by itself a certain number of times, while the exponent is the number that represents how many times the base is multiplied by itself. For example, in the equation 3^4, 3 is the base and 4 is the exponent.

Can an exponent be negative?

Yes, an exponent can be negative. This indicates that the base number should be divided by itself a certain number of times. For example, 3^-2 is equivalent to 1/3^2, which equals 1/9.

How do you solve for an unknown exponent with different bases?

To solve for an unknown exponent with different bases, you can use logarithms to rewrite the equation in a form where the bases are the same. Then, you can solve for the unknown exponent using the laws of exponents or by using the logarithmic properties.

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