Express as a single logarithm and simplify:
2 log 3 + log (3x) – log x
log(3^2)+log(3x) - log x
log (9*3x/x)=log27
To express the expression as a single logarithm and simplify, we can use the properties of logarithms.
The properties we will use are:
1. Product Rule: log(base a) (xy) = log(base a) x + log(base a) y
2. Quotient Rule: log(base a) (x/y) = log(base a) x - log(base a) y
Let's simplify the expression step by step:
2 log 3 + log (3x) – log x
First, let's use the product rule on the first two terms:
log 3^2 + log (3x) – log x
Next, apply the quotient rule to combine the last two terms:
log 3^2 + log (3x/x)
Now, simplify the expression inside the logarithm:
log 9 + log (3)
Finally, use the product rule to combine the two logarithms:
log (9 * 3)
And simplify the result:
log 27
Therefore, the expression 2 log 3 + log (3x) – log x can be expressed as a single logarithm and simplified as log 27.