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. :)

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) ] **

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}.