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To expand logarithms, you can use the following rules:

1. Product rule: The product rule of logarithms states that log base b of (x * y) is equal to the sum of log base b of x and log base b of y. Therefore, to expand a logarithm of a product, you can split it into the sum of logarithms of each factor. For example:

log base 2 of (x * y) = log base 2 of x + log base 2 of y

2. Quotient rule: The quotient rule of logarithms states that log base b of (x / y) is equal to the difference of log base b of x and log base b of y. Therefore, to expand a logarithm of a quotient, you can split it into the difference of logarithms of the numerator and denominator. For example:

log base 2 of (x / y) = log base 2 of x - log base 2 of y

3. Power rule: The power rule of logarithms states that log base b of (x^y) is equal to y times log base b of x. Therefore, to expand a logarithm of a power, you can multiply the exponent by the logarithm of the base. For example:

log base 2 of (x^3) = 3 * log base 2 of x

4. Change of base formula: The change of base formula allows you to change the base of a logarithm. It states that log base b of x can be expressed as log base a of x divided by log base a of b. Therefore, to expand a logarithm with a different base, you can use the change of base formula to express it in terms of a logarithm with a known base. For example:

log base 2 of x = log base 10 of x / log base 10 of 2

It's important to remember that when expanding logarithms, you may need to simplify or combine terms afterwards to get the expression into a more useful or simplified form.