how to wirte 3log(s)-4log(c)+2log(t-4) as a single logarithm. Thanks

To write the expression 3log(s)-4log(c)+2log(t-4) as a single logarithm, we can use the logarithmic properties.

First, let's recall the properties:
1. log(a) + log(b) = log(ab) - This property allows us to combine the addition of two logarithms with the same base into a single logarithm.
2. log(a) - log(b) = log(a/b) - This property allows us to combine the subtraction of two logarithms with the same base into a single logarithm.
3. nlog(a) = log(a^n) - This property allows us to move the coefficient of a logarithm as the exponent of the argument.

Now let's proceed with simplifying the given expression step by step:

1. Start with 3log(s)-4log(c)+2log(t-4).

2. Apply the first property to combine the first two logarithms:
3log(s) - 4log(c) = log(s^3) - log(c^4).

Now the expression becomes log(s^3) - log(c^4) + 2log(t-4).

3. Apply the second property to combine the first two terms:
log(s^3) - log(c^4) = log(s^3/c^4).

Now the expression becomes log(s^3/c^4) + 2log(t-4).

4. We can rewrite 2log(t-4) using the third property as:
2log(t-4) = log((t-4)^2).

Now the expression becomes log(s^3/c^4) + log((t-4)^2).

5. Lastly, apply the first property to combine the two logarithms:
log(s^3/c^4) + log((t-4)^2) = log((s^3/c^4)*(t-4)^2).

So, the final expression as a single logarithm is:
log((s^3/c^4)*(t-4)^2).

Therefore, 3log(s)-4log(c)+2log(t-4) can be written as log((s^3/c^4)*(t-4)^2).