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Question is to solve the equation for value of $x$.

$$9^{1+\log x} - 3^{1+\log x} - 210 = 0; \quad \text{where base of log is }3$$

The answer given is $x=5$

I've tried to solve it. And got two values of $x= -14/3$ and $x=5$. What I've done wrong?

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  • $\begingroup$ Use $a^{\log_b(x)} = x^{\log_b(a)}$. $\endgroup$ – Viktor Glombik Apr 20 '19 at 19:57
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    $\begingroup$ The final step should be $(3k+14)(k-5)=0$ (multiplication since you are factoring) This is what then allows you to reach your conclusion. $\endgroup$ – John Doe Apr 20 '19 at 20:00
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You have solved correctly just made one error towards the end. Note that the domain of $\log(x)$ is $x > 0$ so $x=-14/3$ is rejected as it is not in the domain of the function.

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    $\begingroup$ Yep. And going from $(3 x + 14) + (x - 5) = 0$ does not mean that either term in the sum is $0$. $\endgroup$ – David G. Stork Apr 20 '19 at 19:58
  • $\begingroup$ He meant$(3x+14)(x-5) = 0$. He was solving a quadratic. Must be $\times$ in the middle. $\endgroup$ – Vizag Apr 20 '19 at 20:00
  • $\begingroup$ 😅😅Oh! I didn't noticed that. $\endgroup$ – Piyush Raj Apr 20 '19 at 20:34

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