Write 4logb(X)-1/3logb(y)+2logb(z)as a single logarithm

Its log base b.

4logb(X)-1/3logb(y)+2logb(z)

= logb (x^4) - logb (y(1/3)) + logb (z^2)
= logb (x^4 z^2)/y^(1/3)

To write the expression 4logb(X) - (1/3)logb(y) + 2logb(z) as a single logarithm, we can use the logarithmic property that allows us to combine multiple logarithms with the same base into a single logarithm.

The property states that:

logb(A) + logb(B) = logb(A * B)

Now, let's apply this property step by step:

1. Combine the first two terms: 4logb(X) - (1/3)logb(y)

Using the property stated above:

4logb(X) - (1/3)logb(y) = logb(X^4) - logb(y^(1/3))

2. Now, let's combine the resulting expression with the third term: logb(X^4) - logb(y^(1/3)) + 2logb(z)

Using the property again:

logb(X^4) - logb(y^(1/3)) + 2logb(z) = logb(X^4 * z^2) - logb(y^(1/3))

3. Finally, let's simplify the remaining expression:

logb(X^4 * z^2) - logb(y^(1/3)) = logb((X^4 * z^2) / y^(1/3))

Thus, the expression 4logb(X) - (1/3)logb(y) + 2logb(z) can be written as a single logarithm: logb((X^4 * z^2) / y^(1/3)).

To combine the given logarithmic expression into a single logarithm, we can use the properties of logarithms.

The properties we will use are:
1. logb(X^n) = nlogb(X)
2. logb(X) - logb(Y) = logb(X/Y)

Given: 4logb(X) - 1/3logb(y) + 2logb(z)

Using property 1, we can rewrite the expression as:
logb(X^4) - logb(y^(1/3)) + logb(z^2)

Now, using property 2, we can combine the logarithms with the same base b:
logb(X^4/y^(1/3)) + logb(z^2)

Finally, using property 1 again, we can simplify the expression further:
logb((X^4 * z^2)/y^(1/3))

Therefore, 4logb(X)-1/3logb(y)+2logb(z) can be written as a single logarithm: logb((X^4 * z^2)/y^(1/3))