How do you approximate #log_5 (10/9)# given #log_5 2=0.4307# and #log_5 3=0.6826#?
2 Answers
Approximately
Explanation:
The Logarithmic Division Rule states that:
The Logarithmic Multiplication Rule states that:
We can apply the logarithmic division rule, so:
becomes:
We can simplify the numbers in the brackets into the products of prime numbers:
Now, we can apply the logarithmic multiplication rule, so:
Now, we can substitute in the values given to us:
We apply

The Logarithmic Division Rule which states that
#log(x/y)=logxlogy# 
The Logarithmic Multiplication Rule
#log(x*y)=logx+logy# 
And the Logarithmic Power Rule
#log(x^y)=ylogx#
Given number can be written as
#log_5(10/9)#
#=>log_5((2xx5)/3^2)#
#=>log_5(2xx5)log_5 3^2# .......(Division Rule)
#=>log_5 2+log_5 5log_5 3^2# .......(Multiplication Rule)
#=>log_5 2+log_5 52log_5 3# .......(Power Rule)
Inserting the given values and remembering that
#log_5(10/9)=0.4307+12xx0.6826#
#=>log_5(10/9)=0.0655#