How to solve this logarithmic equation: $ \ x^{3\log^3 x-\big(\frac{2}{3}\big)\log x} = 100 \sqrt[3]{10}\ $

Question

How to solve this?

I am new to logarithms.

$$ \ x^{3\log^3 x-\big(\frac{2}{3}\big)\log x} = 100 \sqrt[3]{10}\ $$ All the logs have base $10$.

Answer 1

It's $$\ (3\log^3{x}-\frac{2}{3}\log{x})\log{x} = 2\frac{1}{3}\ $$ and $\log{x}=t$

We get $$9t^4-2t^2-7=0,$$ which gives $t=1$ or $t=-1$ and the answer is: $$\left\{\frac{1}{10},10\right\}$$

Answer 2

Note that $$\log(100\sqrt[3]{10})=\log(100)+\log(\sqrt[3]{10})$$ and your equation $$(3\lg(x)^3-\frac{2}{3}\lg(x))\lg(x)=\lg(100)+\frac{1}{3}\lg(10)$$

Answer 3

$3log(3)+ (2/3)log(x)= log(27+ x^{2/3}$ so $x^{3log(3)+ (2/3)log(x)}= x^{log(27+ x^{2/3})}$. Further, $100\sqrt[3]{10}= 10^{2+ 1/3}= 10^{7/3}$. So the equation becomes $x^{log(27+ x^{2/3})}= 10^{7/3}$ and taking the logarithm of both sides $(27+ x^{2/3})log(x)= 7/3$