anonymous
  • anonymous
Please help!!!! 3log(x^2) + 4log(2x)=2
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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mathstudent55
  • mathstudent55
CAn you use the rules of logs to come up with a single log expression on the left side?
anonymous
  • anonymous
I know the first couple of steps...
anonymous
  • anonymous
Not sure what you're asking? Sorry.

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mathstudent55
  • mathstudent55
Can you show the steps you already did?
anonymous
  • anonymous
3log(x^2) + 4log(2x)=2 log(x^2)^3 + log(2x)^4=2 log (x^6) + log (16x^4)=2
anonymous
  • anonymous
log (x^24) = 2?
mathstudent55
  • mathstudent55
The last step is not correct, but until the previous step you are correct.
anonymous
  • anonymous
log (x^10) = 2?
mathstudent55
  • mathstudent55
\(3 \log x^2 + 4 \log 2x = 2\) \(\log (x^2)^3 + \log(2x)^4 = 2\) \(\log x^6 + \log (2^4 x^4) = 2\)
mathstudent55
  • mathstudent55
Now we use the rule of logs: \(\log a + \log b = \log (ab) \)
anonymous
  • anonymous
So, log (16x^24) = 2?
mathstudent55
  • mathstudent55
\(\log (x^6 \cdot 2^4x^4) = 2\) \(\log (2^4 x^{10}) = 2\)
anonymous
  • anonymous
oh you add 6 and 4..
mathstudent55
  • mathstudent55
Remember, when you multiply powers with the same base, you ADD the exponents.
mathstudent55
  • mathstudent55
The rule is: \(a^m \cdot a^n = a^{m + n} \)
anonymous
  • anonymous
So for the next step... log(2)= 16x^10?
anonymous
  • anonymous
Right, okay.
mathstudent55
  • mathstudent55
Ok. Now we use the definition of log. Is this log base 10 or natural log?
anonymous
  • anonymous
log base 10
mathstudent55
  • mathstudent55
Definition of log: \(\large \log_b x = y \iff b^y = x\)
mathstudent55
  • 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.
mathstudent55
  • mathstudent55
\(\log (2^4 x^{10}) = 2~~~\iff~~~10^2 = 2^4x^{10}\) Do you understand this step?
anonymous
  • anonymous
Ah, yes
mathstudent55
  • mathstudent55
Great. Now let's continue solving for x.
anonymous
  • anonymous
100=16x^10 Divide 16 6.25=x^10 6.25^(1/10) x=1.2011
mathstudent55
  • mathstudent55
\(2^4x^{10} = 10^2\) \(x^{10} = \dfrac{10^2}{2^4} \) \(x^{10} = \left( \dfrac{10}{2^2} \right)^2\) \(x^{10} = \left( \dfrac{5}{2} \right)^2\) \(\left (x^{10} \right)^{\frac{1}{10}} = \left( \left( \dfrac{5}{ 2} \right)^2\ \right)^{\frac{1}{10}} \)
anonymous
  • anonymous
I understand now, thanks so much!
mathstudent55
  • mathstudent55
\(x = \left( \dfrac{5}{2} \right) ^ {\frac{1}{5}} \) \(x = \sqrt[5] {\dfrac{5}{2} } \)
mathstudent55
  • mathstudent55
You're welcome.

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