How Do You Multiply Logs with Different Bases?

To multiply logarithms with different bases, you can use the change of base formula. The change of base formula states that for any logarithm, you can convert it to a different base by using the formula:

log_b(a) = log_c(a) / log_c(b)

where c is the new base you want to convert to.

When multiplying two logarithms, say log_a(b) and log_c(d), you can apply the change of base to convert both logarithms to the same base. For example, if you want to convert them to base 10, you would get:

• log_a(b) = log₁₀(b) / log₁₀(a)
• log_c(d) = log₁₀(d) / log₁₀(c)

Now, you can multiply these two expressions:

log_a(b) * log_c(d) = (log₁₀(b) / log₁₀(a)) * (log₁₀(d) / log₁₀(c))

This will give you the product of the two logarithms in terms of base 10 logarithms. You can simplify further if necessary. This process allows you to multiply logs with different bases by converting them to a common base before applying the multiplication.