Question 814157
Here's a procedure for solving these types of problems where they give you the value of some logarithms and ask you to find the value of another log (without using a calculator):<ol><li>Rewrite the argument of the desired log as a product, quotient and/or powers of the numbers whose logs you know. Note: In addition to the logs you're given you should also know that {{{log(a, (a)) = 1}}}. So, since we're using base 10 logs in this problem, we should know that log(10) = 1 without it being given to you. So in this problem we start knowing the base 10 logs of 2, 3 and 10.</li><li>Use properties of logarithms to rewrite the expression in terms of log's of 2's, 3's and/or 10's.</li><li>Substitute in the known values for these logs and simplify.</li></ol>So for the first step we will want to rewrite 1/9 as a product, quotient and/or power of 2's, 3's and/or 10's. I hope it doesn't take long for you to see that 9 is a power of 3. But what about 1/9? If you know your negative exponents then you will see that {{{1/9 = 3^(-2)}}}. So
{{{log((1/9))}}}
can be rewritten as:
{{{log((3^(-2)))}}}
Now we need to rewrite this so that it is in terms of log(3). So the -2 has to move. For this we use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}
{{{(-2)log((3))}}}
And finally we substitute in the known value for log(3):
{{{(-2)* 0.4771}}}
which simplifies to:
-0.9542