- #1

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a^x + x^b = c

Where x is the unknown (real or complex) and a, b, c real constants.

Or at least if you know a line of attack.

- Thread starter Aphex_Twin
- Start date

- #1

- 39

- 0

a^x + x^b = c

Where x is the unknown (real or complex) and a, b, c real constants.

Or at least if you know a line of attack.

- #2

- 208

- 0

u can use the property of logarithma

x = log a

base b

b^x = a

x = log a

base b

b^x = a

- #3

- 39

- 0

a^x + x^b = c

x^x = c - x^b

x*ln(a) = ln(c - x^b)

x = ln(c-x^b)/(ln(a))

x = log_a (c-x^b)

??

where log_a is the base-a logarithm

- #4

HallsofIvy

Science Advisor

Homework Helper

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Other than that, a numerical solution such as Newton's method.

- #5

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I start with a simpler equation, namely: x=2*ln(x)

e^x = x^2

1 = x^2/e^x

1 = x^2 * e^(-x)

1 = x * e^(-x/2)

-1/2 = -x/2 * e^(-x/2)

Therefore x = W(-1/2)

So going ahead with the main equation:

x = ln(c-x^b) * 1/(ln(a))

1 = ln(c-x^b) * (1/ln(a)) * 1/x

ln(a) = ln (c - x^b) * 1/x

a = (c - x^b) * e^(1/x)

I am stuck again

- #6

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- 0

a^x+x^b=c

a^x = c - x^b

x*ln(a) = ln(c - x^b)

x = ln(c-x^b)/(ln(a))

x = ln(c-x^b) * (1/(ln(a)))

1 = ln(c-x^b) * (1/ln(a)) * 1/x

ln(a) = ln (c - x^b) * 1/x

ln(a) = ln(c-x^n)*e^(-ln(x))

(ln(a))^n = (ln(c-x^n))^n*e^(-ln(x^n))

(ln(a)^(n*ln(-1)) = (ln(c-x^n))^(n*ln(-1))*e^(-ln(-x^n))

(ln(a)^(n*ln(-1)*c) = ln(n*ln(-1)*c)*(c-x^n)*e^(-ln(c-x^n))

(ln(a))^(n*c*pi*i)=ln(n*c*pi*i)*(c-x^n)*e^(-ln(c-x^n))

ln(a) = ln(c-x^n)*e^((-ln(c-x^n)+1/(n*c*pi*i))

ln(a)=ln(c-x^n)*e^(1/(n*c*pi*i))*e^(-ln(c-x^n))

-ln(a)/(e^(1/(n*c*pi*i))) = -ln(c-x^n)*e^(-ln(c-x^n))

-ln(c-x^n) = W(-ln(a)/(e^(1/(n*c*pi*i))))

c-x^n = -e^(W(-ln(a)/(e^(1/(n*c*pi*i)))))

x^n = e^(W(-ln(a)/(e^(1/(n*c*pi*i))))) - c

x = (e^(W(-ln(a)/(e^(1/(n*c*pi*i))))) - c)^(1/n)

Did I lose a sign or something on the way? :yuck:

- #7

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- 0

whoops, I miswrote b as n somewhere on the mid way.

a^x+x^b=c

a^x = c - x^b

x*ln(a) = ln(c - x^b)

x = ln(c-x^b)/(ln(a))

x = ln(c-x^b) * (1/(ln(a)))

1 = ln(c-x^b) * (1/ln(a)) * 1/x

ln(a) = ln (c - x^b) * 1/x

ln(a) = ln(c-x^b)*e^(-ln(x))

(ln(a))^b = (ln(c-x^b))^b*e^(-ln(x^b))

(ln(a)^(b*ln(-1)) = (ln(c-x^b))^(b*ln(-1))*e^(-ln(-x^b))

(ln(a)^(b*ln(-1)*c) = ln(b*ln(-1)*c)*(c-x^b)*e^(-ln(c-x^b))

(ln(a))^(b*c*pi*i)=ln(b*c*pi*i)*(c-x^b)*e^(-ln(c-x^b))

ln(a) = ln(c-x^b)*e^((-ln(c-x^b)+1/(b*c*pi*i))

ln(a)=ln(c-x^b)*e^(1/(b*c*pi*i))*e^(-ln(c-x^b))

-ln(a)/(e^(1/(b*c*pi*i))) = -ln(c-x^b)*e^(-ln(c-x^b))

-ln(c-x^b) = W(-ln(a)/(e^(1/(b*c*pi*i))))

c-x^b = -e^(W(-ln(a)/(e^(1/(b*c*pi*i)))))

x^b = e^(W(-ln(a)/(e^(1/(b*c*pi*i))))) - c

x = (e^(W(-ln(a)/(e^(1/(b*c*pi*i))))) - c)^(1/b)

a^x+x^b=c

a^x = c - x^b

x*ln(a) = ln(c - x^b)

x = ln(c-x^b)/(ln(a))

x = ln(c-x^b) * (1/(ln(a)))

1 = ln(c-x^b) * (1/ln(a)) * 1/x

ln(a) = ln (c - x^b) * 1/x

ln(a) = ln(c-x^b)*e^(-ln(x))

(ln(a))^b = (ln(c-x^b))^b*e^(-ln(x^b))

(ln(a)^(b*ln(-1)) = (ln(c-x^b))^(b*ln(-1))*e^(-ln(-x^b))

(ln(a)^(b*ln(-1)*c) = ln(b*ln(-1)*c)*(c-x^b)*e^(-ln(c-x^b))

(ln(a))^(b*c*pi*i)=ln(b*c*pi*i)*(c-x^b)*e^(-ln(c-x^b))

ln(a) = ln(c-x^b)*e^((-ln(c-x^b)+1/(b*c*pi*i))

ln(a)=ln(c-x^b)*e^(1/(b*c*pi*i))*e^(-ln(c-x^b))

-ln(a)/(e^(1/(b*c*pi*i))) = -ln(c-x^b)*e^(-ln(c-x^b))

-ln(c-x^b) = W(-ln(a)/(e^(1/(b*c*pi*i))))

c-x^b = -e^(W(-ln(a)/(e^(1/(b*c*pi*i)))))

x^b = e^(W(-ln(a)/(e^(1/(b*c*pi*i))))) - c

x = (e^(W(-ln(a)/(e^(1/(b*c*pi*i))))) - c)^(1/b)

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