## anonymous one year ago Please help!!!! 3log(x^2) + 4log(2x)=2

1. mathstudent55

CAn you use the rules of logs to come up with a single log expression on the left side?

2. anonymous

I know the first couple of steps...

3. anonymous

Not sure what you're asking? Sorry.

4. mathstudent55

Can you show the steps you already did?

5. anonymous

3log(x^2) + 4log(2x)=2 log(x^2)^3 + log(2x)^4=2 log (x^6) + log (16x^4)=2

6. anonymous

log (x^24) = 2?

7. mathstudent55

The last step is not correct, but until the previous step you are correct.

8. anonymous

log (x^10) = 2?

9. mathstudent55

$$3 \log x^2 + 4 \log 2x = 2$$ $$\log (x^2)^3 + \log(2x)^4 = 2$$ $$\log x^6 + \log (2^4 x^4) = 2$$

10. mathstudent55

Now we use the rule of logs: $$\log a + \log b = \log (ab)$$

11. anonymous

So, log (16x^24) = 2?

12. mathstudent55

$$\log (x^6 \cdot 2^4x^4) = 2$$ $$\log (2^4 x^{10}) = 2$$

13. anonymous

oh you add 6 and 4..

14. mathstudent55

Remember, when you multiply powers with the same base, you ADD the exponents.

15. mathstudent55

The rule is: $$a^m \cdot a^n = a^{m + n}$$

16. anonymous

So for the next step... log(2)= 16x^10?

17. anonymous

Right, okay.

18. mathstudent55

Ok. Now we use the definition of log. Is this log base 10 or natural log?

19. anonymous

log base 10

20. mathstudent55

Definition of log: $$\large \log_b x = y \iff b^y = x$$

21. mathstudent55

Now let's use the last equation we have as the left side of the definition of log, and change it into the exponential version on the right side.

22. mathstudent55

$$\log (2^4 x^{10}) = 2~~~\iff~~~10^2 = 2^4x^{10}$$ Do you understand this step?

23. anonymous

Ah, yes

24. mathstudent55

Great. Now let's continue solving for x.

25. anonymous

100=16x^10 Divide 16 6.25=x^10 6.25^(1/10) x=1.2011

26. mathstudent55

$$2^4x^{10} = 10^2$$ $$x^{10} = \dfrac{10^2}{2^4}$$ $$x^{10} = \left( \dfrac{10}{2^2} \right)^2$$ $$x^{10} = \left( \dfrac{5}{2} \right)^2$$ $$\left (x^{10} \right)^{\frac{1}{10}} = \left( \left( \dfrac{5}{ 2} \right)^2\ \right)^{\frac{1}{10}}$$

27. anonymous

I understand now, thanks so much!

28. mathstudent55

$$x = \left( \dfrac{5}{2} \right) ^ {\frac{1}{5}}$$ $$x = \sqrt[5] {\dfrac{5}{2} }$$

29. mathstudent55

You're welcome.