How do you evaluate this log equation?

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The discussion revolves around evaluating the logarithmic equation log39 + log41/64 + log51. Initial attempts to solve it involved incorrect interpretations of the logarithmic properties, leading to inaccurate results. Participants clarified that the correct approach involves calculating each log expression individually rather than combining them as equations. The final solution is derived as log39 + log41/64 + log51 = 2 - log464 + 0, simplifying to -1. Proper understanding of logarithmic identities is emphasized for accurate results.
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Homework Statement



log39 + log41/64 + log51


Homework Equations


none


The Attempt at a Solution



I tried answering it by...

(3x=9) + (4x=1/64) + (5x=1)
because of this, I got 5. Is that correct?
 
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Hi haengbon! :smile:
haengbon said:
(3x=9) + (4x=1/64) + (5x=1)
because of this, I got 5. Is that correct?

Correct method :smile:, but wrong result :redface: …

try (4x=1/64) again. :wink:​
 
tiny-tim said:
Hi haengbon! :smile:


Correct method :smile:, but wrong result :redface: …

try (4x=1/64) again. :wink:​

I tried it again and got -3 for 4x=1/64 :D is that correct now? ^^ thank you for the help by the way :)
 
haengbon said:
I tried it again and got -3 for 4x=1/64 :D is that correct now? ^^

Yup! :biggrin:
 
haengbon said:
I tried answering it by...

(3x=9) + (4x=1/64) + (5x=1)
because of this, I got 5. Is that correct?
If you turn in work like this, your instructor will probably deduct points, since you are apparently trying to add three equations.

Each of the log expressions is simple enough to calculate, so your work should look something like this.

log39 + log41/64 + log51
= 2 - log464 + 0
= 2 - 3 = -1

I used the fact that log(1/x) = - log(x) going from the first expression to the second.
 

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Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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