How do you expand or condense a logarithm?

Category: Logarithms »

Relevant MM Lessons:

Overview: Expanding and Collapsing Logarithms

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how expanding and condensing log expressions each shows up on tests.Here are the short notes on each log process - check out the full lesson for extra background and understanding.

Expanding Logarithms

When we're asked to expand a logarithm, we will be given a single log expression and are expected to write it as the sum and/or difference of several simpler logarithms. What that comes down to for us is that every factor in the numerator of the log argument will be its own positive logarithm, and every factor in the denominator of the log argument will be its own negative logarithm. If any factor has an exponent, that exponent will be placed in front of the corresponding log as a multiplier.For example:$$\log \left( \frac{x^2 y}{w^3 z^5} \right)$$$$=2 \log (x) + \log(y) - 3\log (w) - 5 \log(z)$$Make sure every log has only a single, simple object in it when you're done. If you want to see more difficult problems or better understand why this end result always happens, read the dedicated lesson » on expanding logs.

Condensing Logarithms

When we're asked to condense a logarithmic expression, we are essentially doing the opposite of what we just saw for expanding. The problems we'll see will have the sum or difference of several logs, with and without coefficients in front. Our job is to first bring any coefficients inside the logarithms, and then to compile the pieces into a single logarithm with a single argument. Similar to the concepts involved with expanding logs, positive logs will turn into numerator items, while negative logs will end up in the denominator. It is possible to become masterful at this and crush problems flawlessly with one step, but I recommend two for clarity.For example:$$4\log(a+b) - \log(x) - 3\log(c) + 2\log(w) - \log (y-z) + 2\log (4)$$I recommend quickly writing things with no coefficients. Then you'll have each piece ready to place.$$\log\big[ (a+b)^4 \big] - \log(x) - \log\big(c^3 \big) +\log\big(w^2\big) - \log (y-z) + \log (16)$$$$\longrightarrow \log \left( \frac{16(a+b)^4 w^2}{c^3 x(y-z)} \right)$$Note that for condensing logs, all the log expressions must be the same log base - otherwise you cannot combine them! There are ways teachers can try and trick you with this stuff - again refer to the dedicated lesson » for more tips and a more thorough explanation.