if: $\log_{a}{ax}+2\log_{a^2}{ax}+3\log_{a^3}{ax}...+10\log_{a}{a^{10}x}$ what is x?

Question

if: $\log_{a}{ax}+2\log_{a^2}{ax}+3\log_{a^3}{ax}...+10\log_{a}{a^{10}x}=110$ what is x?

I've simplified the equation to: $\log_{a}{a^{385}x^{55}}=110$

but then I have no clue what to do afterwards

Answer 1

From $\log_{a}(a^{385}x^{55})=110$:

$$\log_{a}(a^{385})+\log_{a}(x^{55})=110$$ $$385+\log_{a}(x^{55})=110$$ $$\log_{a}(x^{55})=-275$$ $$x^{55}=a^{-275}$$ $$(x^{55})^{\frac{1}{55}}=(a^{-275})^{\frac{1}{55}}=x^{55 \cdot \frac{1}{55}}=a^{-275 \cdot \frac{1}{55}}=x=a^{-\frac{275}{55}}=a^{-5}$$

Therefore, we have:

$$x=a^{-5}$$