Question 235481: 4log4 6 - log 4 5, evaluate without a calcuator
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 4log4 6 - log 4 5, evaluate without a calculator
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log4 [(6^4)/5] = 4.00896...
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I can't imagine how you would get
that answer without a calculator.
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Cheers,
Stan H.
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
The best I can do without a calculator is to express this in terms of a single logarithm.
Use
 = n\log_b(x))
To write:
\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5))
Then use: The difference of the logs is the log of the quotient, that is:
 - \log_b(y) = log_b\left(\frac{x}{y}\right))
to write:
\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5)\ =\ \log_4\left(\frac{6^4}{5}\right))
I need at least a scratch pad to go any further:  ,  , 
\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5)\ =\ \log_4\left(\frac{6^4}{5}\right)\ =\ \log_4(259.2))
Now, if you want to do this the old fashioned way with a table of logarithms, you aren't likely to find a table of base 4 logs. That means you will have to do a base conversion.
Here's the formula:
\ =\ \frac{\log_c(x)}{\log_c(b)})
So, for your situation:
\ =\ \frac{\log_{10}(259.2)}{\log_{10}(4)})
Then you would have to look up ) and ) , then interpolate to find ) .
To interpolate: \ + .2\left(\log_{10}(2.60)\ - \log_{10}(2.59)\right)) .
Finally, calculate \ +\ 2\ =\ \log_{10}(259.2)) .
Next, look up ) and divide it into the results of the last step -- by hand, of course.
Good luck.
John
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