Solving $9^{1+\log x} - 3^{1+\log x} - 210 = 0$ where base of log is $3$ for $x$

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?

• Use $a^{\log_b(x)} = x^{\log_b(a)}$. Apr 20 '19 at 19:57
• 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. Apr 20 '19 at 20:00

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.
• Yep. And going from $(3 x + 14) + (x - 5) = 0$ does not mean that either term in the sum is $0$. Apr 20 '19 at 19:58
• He meant$(3x+14)(x-5) = 0$. He was solving a quadratic. Must be $\times$ in the middle. Apr 20 '19 at 20:00