Question 269655
Taking an expression of multiple logarithms as a single logarithm requires that we know the properties of logarithms:<ul><li>{{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}</li><li>{{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}</li><li>{{{q*log(a, (p)) = log(a, (p^q))}}}</li></ul>
The first two are used to combine two logarithms into one. The first one is use when there is a "+" between the two logs. The second one is used when there is a "-" between the two logs. The first two properties require that the two logarithms have coefficients of 1 in front of them. The third property is used to move other coefficients into the argument of the logarithm. Let's see how this works on your expression:
{{{log(b, (5x)) + 7(log(b, (x)) - log(b, (y)))}}}
We'll start by combining the two logarithms in the parentheses using the second property:
{{{log(b, (5x)) + 7(log(b, (x/y)))}}}
or
{{{log(b, (5x)) + 7*log(b, (x/y))}}}
Before we combine the last two logarithms we need to have coefficients of 1. So that 7 needs to go. This is where the thrid property comes in handy:
{{{log(b, (5x)) + log(b, ((x/y)^7))}}}
or
{{{log(b, (5x)) + log(b, (x^7/y^7))}}}
Now we can combine the last two logarithms (with the first property):
{{{log(b, (5x*(x^7/y^7)))}}}
which simplifies to
{{{log(b, (5x^8/y^7))}}}