## mathmath333 one year ago logarithm question

1. mathmath333

solve \large \color{black}{\begin{align} \log_{10}(x^{3}+5)=3\log_{10}(x+2)\hspace{.33em}\\~\\ \end{align}}

2. mathmath333

\large \color{black}{\begin{align} & (a.)\ \dfrac{-2+\sqrt2}{2} \hspace{.33em}\\~\\ & (b.)\ \dfrac{-2-\sqrt2}{2} \hspace{.33em}\\~\\ & (c.)\ \text{both a.) and b.) } \hspace{.33em}\\~\\ & (d.)\ \text{none of these} \hspace{.33em}\\~\\ \end{align}}

3. ParthKohli

$\log_{10}(x^3 + 5) = \log_{10}{(x+2)^3}$$\Rightarrow x^3 + 5 = x^3 + 6x^2 + 12x + 8$$\Rightarrow 6x^2 + 12x + 3 = 0$Choose the solution for which both logarithms are defined.

4. mathmath333

both a.) and b.)

5. mathmath333

?

6. ParthKohli

Yeah, I guess so.

7. mathmath333

my book has given option d.)

8. ParthKohli

9. mathmath333

lol i am not sure

10. mathmath333

i dont have brains

11. ParthKohli

You're correct. Don't worry.

12. mathmath333

all options of book is correct except this ? suspictious

13. ParthKohli

Book bhi ek insaan ne hi likhi hai. As I said, your answer is correct.

14. ganeshie8

you may double check with wolfram

15. mathmath333
16. ParthKohli

So you're right.

17. mathmath333

ok thnx, by the way is there a condition like $$x^{3}+5>0$$ and $$(x+2)^{3}>0$$

18. ParthKohli

Yes, which is what I meant that the logarithms should be defined. Both roots satisfy both conditions.

19. mathmath333

how do i find value of $$5^{1/3}$$ by pen paper to check

20. ParthKohli

You don't need to find the value of $$5^{1/3}$$. Plug in $$1.4$$ for $$\sqrt 2$$ and that should almost always work.

21. mathmath333

y i dont need that value

22. ParthKohli

You need to check if $$x^3 + 5 > 0$$ and $$x+2>0$$. The second one is easier than the first. The solutions you get are$\frac{-2 + \sqrt 2}{2} \approx - 0.3 , \frac{-2 - \sqrt 2}{2} \approx -1.7$The second one is simple. The first condition, well, is satisfied the first root. The second root is a closer call but it satisfies the condition.

23. ParthKohli

satisfied for the first root*

24. phi

You are the victim of a typo.... it happens quite a bit in math books because it is very hard to check the answers....

25. mathmath333

no typo for x^3+5>0

26. ParthKohli