How do you write the following expression as a single logarithm: log 5 - log x - log y?

1 Answer
Pradyun Devarakonda
May 23, 2018

#log(5/(xy))#

Explanation:

We have two properties in Logarithms that can be used in solving this question

  • #log_m(a) + log_m(b) = log_m(a*b)#

proof :-

Let :

# log_m(a) = p => m^p = a# ------1

and

#log_m(b) = q => m^q = b# ------2

So,

From 1 and 2 ;

#a*b = m^p * m^q#

#:. a * b = m^(p+q)#

By writing #uarr# in logarithmic form

#log_m(ab) = p + q#

#=> log_m(ab) = log_m(a) + log_m(b) #

and

  • #log_m(a) - log_m(b) = log_m(a/b)#

proof:-

Let :

#log_m(a) = p => m^p = a# -------1

and

#log_m(b) = q => m^q = b# ------2

So,

From 1 and 2 ;

#a/b = m^p / m^q#

#:. a / b = m^(p-q)#

By writing #uarr# in logarithmic form

#log_m(a/b) = p - q#

#=> log_m(a/b) = log_m(a) - log_m(b) #

The given question is :

#log(5) - log (x) - log(y)#

#=> log(5) - ( log(x) + log(y) )#

#=> log(5) - ( log(x*y) )#

#=> log(5/(x*y) )#

#=> log(5/(xy) )#

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