Question 678074
The "trick" with these problems is to use properties of logarithms to rewrite the desired log in terms of the logs you've been given. So in this problem we will use properties to transform {{{log(b, (32/125))}}} into an expression involving logs of 2, 3 and/or 5.<br>
With a little effort we should be able to figure out that {{{32 = 2^5}}} and {{{125 = 5^3}}}. So {{{log(b, (32/125))}}} can be written as:
{{{log(b, (2^5/5^3))}}}
Now we can start using properties to rewrite this in terms of logs of 2 and 5. First we use the property {{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}} to split the numerator and denominator into separate logs:
{{{log(b, (2^5)) - log(b, (5^3))}}}
Next we can use the property {{{log(a, (p^n)) = n*log(a, (p))}}} to move the exponents out of the arguments:
{{{5*log(b, (2)) - 3*log(b, (5))}}}
We now have {{{log(b, (32/125))}}} expressed in terms of logs of 2 and 5. All we have to do now is substitute in the given values for these logs:
{{{5*(0.3562) - 3*(0.8271)}}}
Simplifying...
{{{1.781 - 2.4813}}}
{{{-0.7003}}}