A possible value of x, for which the ninth term in the expansion of (3^_3√5^x-1+7+3^-18_3(…

Question

A possible value of $x$, for which the ninth term in the expansion of $\left(3^{\log_3\sqrt{5^{x-1}+7}}+3^{-\frac18\log_3(5^{x-1}+1)}\right)^{10}$ is equal to 180, is

Accepted Answer

Simplify the bases: $3^{\log_3\sqrt{5^{x-1}+7}}=\sqrt{5^{x-1}+7}$, and $3^{-\frac18\log_3(5^{x-1}+1)}=(5^{x-1}+1)^{-1/8}$.

$$T_9=\binom{10}{8}\left(\sqrt{5^{x-1}+7}\right)^2\left((5^{x-1}+1)^{-1/8}\right)^8=45\left(5^{x-1}+7\right)(5^{x-1}+1)^{-1}=180$$

Let $t=5^{x-1}$: $\dfrac{t+7}{t+1}=4\Rightarrow t+7=4t+4\Rightarrow t=1$

$$5^{x-1}=1\Rightarrow x-1=0\Rightarrow x=1$$

Solutions · 1