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Find the exact solution of $$f(x)=0$$ where $$f(x) = 3^{3x+1}-7\cdot 5^{2x}$$

In my Numerical Analysis class, we have used methods such as Newton's method, Secant method, and bisection method to approximate the solutions of root finding problems. However, we have not discussed any methods to find an exact root.

What methods can I use to solve the above problem?

Also, as a side question, is it possible to (after many iterations) to find an exact root, with one of the approximation methods mentioned above (or others not mentioned)?

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from $$3^{3x+1}=7\cdot 5^{2x}$$ we get $$(3x+1)\ln(3)=\ln(7)+2x\ln(5)$$ con you proceed?

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  • $\begingroup$ taking the logarithm on both sides $\endgroup$ – Dr. Sonnhard Graubner Mar 8 '16 at 3:48
  • $\begingroup$ I had an error in my original equation. There shouldn't have been a second =, it should have been a -. Does this change the answer? $\endgroup$ – Matt C Mar 8 '16 at 4:17
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    $\begingroup$ No, $a-b=0$ is equivalent to $a=b$. $\endgroup$ – David Mar 8 '16 at 4:43

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