# simplify the followingL 2 log5 a - 3log b + log ( a+ b)

changchengliang | Certified Educator

First of all, I must say that if you had intended for the first term  "2log5 a" to mean "2 times log to base 5" and the argument being "a", then please go right to the last section of this answer: Interpretation 2.

Else, my interpretation of your question is:

Interpretation 1

2.log(5a) - 3.log(b) + log(a+b)

ie. All log functions in base 10.

2.log(5a) - 3.log(b) + log(a+b)

=  log(5a)^2 - log(b^3) + log(a+b)

log[ (25a^2)(a+b) /b^3 ]

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Interpretation 2

2.log5 (a) - 3.log(b) + log(a+b)

= 2.log5 (a) - log(b^3) + log(a+b)

= log5 (a^2) + log[ (a+b) / b^3 ]

= [log(a^2)]/[log 5]  + log[ (a+b) / b^3 ]

= log{ a^[2/(log 5)]} + log[ (a+b) / b^3 ]

= log { a^[2/(log 5)] . (a+b) / b^3 }

What a complicated expression, isn't it?  This leaves us to wonder if this is the original intent of the question.  :)

hala718 | Certified Educator

Let E = 2 log5 a - 3log b + log (a+b)

Using logarethim properties we  know that:

a*log b = log b^a

==> E = log5 (a^2) - log b^2 + log (a+ b)

Now we will rewrtie log5 (a^2) to the base 10:

We know that:

log a b = log c b / logc a

==> log5 a^2 = log a^2 / log 5

==> E = log a^2 / log 5 - log b^2 + log (a+ b)

We know that:

log a + log b = log a / log b

= log a^2 / log 5 - [ log b^2 ( a+ b) ]

neela | Student

To simplify 2log5 a -3logb +log(a+b).

We see that  the second and 3rd terms logarithms to the  base 10. Whereas the  first term 2lo5 (a) is the logarithm  to the base 5.

Therefore before simplification we convert the first term2log5(a)  also to the base 10.

We know by property of lagarithms that log x (y) = log z (y)/ log z(x)).

2log 5(a) = log (a^2) / log5 = log a^(2/log5), as mlog = loga^m.

Therefore  2log5 (a) - 3logb+log(a+b) = log a^(/log5) - log^3+log(a+b) = log {(a+b)[ a^(2/log5)]/b}

Therefore  2 log5 a - 3log b + log ( a+ b) = log {(a+b)[ a^(2/log5)]/b}.