Write expression as one logarithm and simplify if appropriate.

log 3√x + log x^4 - log x^3

4 log (x+3) - 5 log (x^2+4) + 1/3 log y

I have these who problems but I don't know where to start. HELP Please.

You are going to use 3 man rules of logs

1. log(A/B) = logA - logB
2. Log(AB) = logA + logB
3. log A^n = nlogA

log 3√x + log x^4 - log x^3
= log 3 + log x^(1/2) + 4logx - 3logx
= log3 + (1/2)logs + logx
= log3 + (3/2)logx

2.

4 log (x+3) - 5 log (x^2+4) + 1/3 log y
= log (x+3)^4 - log (x^2+4)^5 + log y^(1/3)

= log [ y^(1/3) (x+3)^4 /(x^2+4)^5 ]

To simplify the given expressions and write them as a single logarithm, we will use the properties of logarithms:

1. For the first expression, we have:
log 3√x + log x^4 - log x^3

Using the property log a + log b = log (ab), we can combine the first two terms:
= log (3√x * x^4) - log x^3

Simplifying the expression inside the first logarithm:
= log (3√(x * x * x * x)) - log x^3

Using the property log a^b = b * log a, we can simplify further:
= log (3 * √(x^4)) - log x^3

Simplifying the expression inside the first logarithm:
= log (3 * x^2) - log x^3

Now, using the property log a - log b = log (a/b), we can combine the two logarithms:
= log (3x^2 / x^3)

Simplifying the fraction inside the logarithm by dividing like terms:
= log (3/x)

Therefore, the simplified expression is: log (3/x).

2. For the second expression, we have:
4 log (x+3) - 5 log (x^2+4) + 1/3 log y

Using the property log a^b = b * log a, we can simplify each logarithm:
= log ((x+3)^4) - log ((x^2+4)^5) + log y^(1/3)

Now, using the properties log a - log b = log (a/b) and log a + log b = log (ab) as required, we can combine the logarithms:
= log ((x+3)^4 / (x^2+4)^5) + log y^(1/3)

Simplifying the fraction inside the first logarithm:
= log (((x+3)/(x^2+4))^4) + log y^(1/3)

Simplifying the exponent of y using the property log a^b = b * log a:
= log (((x+3)/(x^2+4))^4) + (1/3) log y

Therefore, the simplified expression is: log (((x+3)/(x^2+4))^4) + (1/3) log y.

By following these steps, you can simplify and combine logarithmic expressions into a single logarithm.