Which statements are true?

a) `log_(a)b-log_(a)c=(log_(a)b)/(log_(a)c)`

b) `log_(a)b-log_(a)c=log_(a)(b/c)`

c) `log_(4)(5x+16)=log_(4)5x+log_(4)16=2+log_(4)5x`

d) `log_(4)(16 xx 5x)=log_(4)16+log_(4)5x=2+log_(4)5x`

e) `log_(a)3x^2=2log_(a)3+2log_(a)x`

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According to the quotient rule of logarithm:

`log_ab-log_ac=log_a(b/c)`

Hence, statement b) is true and statement a) is false.

Statement c) is false since there is no such rule for the split up.

According to the product rule of logarithm:

`log_a(b*c)=log_ab+log_ac`

So, `log_4(16*5x)=log_4(16)+log_4(5x)`

`=log_4(4^2)+log_4(5x)`

Using the exponent rule of logarithm i.e `log_am^n=nlog_am`

`2log_4(4)+log_4(5x)`

`=2+log_4(5x)`

Hence, statement d) is true.

Again using the product and the exponent rule of logarithm,

`log_a(3*x^2)`

`=log_a3+log_ax^2`

`=log_a3+2log_ax`

Hence, statement e) is false.

Therefore, statements **b)** and** d)** are true.

**Sources:**

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