SOLUTION: \(\log _{8}(x+2)-\log _{8}(x)=\log _{8}(30)\)

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Question 1206182: \(\log _{8}(x+2)-\log _{8}(x)=\log _{8}(30)\)
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52794)   (Show Source): You can put this solution on YOUR website!
.

I read it this way

     -  = .


If so, then, using the standard logarithm properties

     = 30,

    x+2 = 30x

    2 = 30x - x

    2 = 29x

    x = .    ANSWER

Solved.

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On logarithms and their properties,  see these introductory lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Evaluate logarithms without using a calculator
    - Simplifying expressions with logarithms
    - Solving logarithmic equations
in this site.



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your problem statement is:
log8(x+2) - log8(x) = log8(30)
since log(a) - log(b) = log(a/b), you get:
log8((x+2)/x) = log8(30)
this is true if and only if (x+2)/x = 30
multiply both sides of that equation by x to get x+2 = 30x.
subtract x from both sides of that equati0on to get 2 = 29x.
solve for x to get x = 2/29.
that should be your answer.
to confirm, replace x with 2/29 in the original equation and solve.
log8(x+2) - log8(x) = log8(30) becomes log8(2/29 + 2) - log8(2/29) = log8(30).
by the log base conversion formula that says log8 = log/log(8), you get:
log(2/29 + 2)/log(8) - log(2/29)/log(8) = log(30)/log(8).
use your calculator to get 1.635630199 = 1.635630199, confirming the equation is true when x = 2/29.

the log function on your calculator is log10 which translates to log to the base of 10.
the log base conversion formula says that log8(x) = log10(x)/log10(8).
since log10 is the log function on your calculator, this becomes log8(x) = log(x)/log(8).

your solution is x = 2/29.





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