How would you condense a logarithm expression into a single quantity if the logs have different bases?
not much you can do
log_3(27) + log_2(64) really have nothing in common
One thing you can do is convert to a common base:
log_3(27) + log_3(64)/log_3(2)
= log_3(27) + log_3(64^1/log^3(2))
= log_3[27 * 64^1/log_3(2)]
but that's not too simple
To condense a logarithm expression with different bases into a single quantity, you can use the change of base formula and rewrite each logarithm in a common base. Here are the steps to follow:
1. Start with the logarithm expression containing different bases, for example: log(base a)(x) + log(base b)(y).
2. Choose a common base for the logarithms. It can be any positive number except 1. Let's say we want to rewrite the expression in base c.
3. Use the change of base formula, which states: log(base a)(x) = log(base c)(x) / log(base c)(a). Similarly, log(base b)(y) = log(base c)(y) / log(base c)(b).
4. Apply the change of base formula to each logarithm in the expression. We get: log(base c)(x) / log(base c)(a) + log(base c)(y) / log(base c)(b).
5. Combine the terms with a common denominator: (log(base c)(x) * log(base c)(b) + log(base c)(y) * log(base c)(a)) / (log(base c)(a) * log(base c)(b)).
6. Finally, simplify the expression if possible.
Note that the base c you choose can be any base, but it is typically chosen to be 10 (common logarithm) or e (natural logarithm).