in this lesson instead of expanding a single log into multiple logarithms we're going to condense multiple logs into a single log so this is the opposite of the last lesson so hopefully you've watched the last lesson before this one so we're going to write this as a single log now the ones that contain a positive sign will go on top so that's a and c so it's going to be a times c on top now b has a negative in front of it so it's going to go on the bottom so it's going to be log ac over b let's try another example 2 log a plus 5 log b minus 7 log c so how can we express this expression as a single log the first thing we need to do is move the coefficient to the exponent position so this is going to be log a squared plus log b to the fifth minus log c to the seventh now we can write a single log a and b both have a positive sign in front so they're going to be on top a squared b to the fifth c has a negative sign in front of it so c to the seventh is going to be on the bottom therefore this is the answer try this one one third log a minus two thirds log b plus one fourth log c minus four over five log d so first so let's move all of the coefficients to the exponent position so this is equal to log a raised to the 1 3 minus log b to the two thirds plus log c to the one fourth minus log d to the four fifths and now let's write it as a single log so a and c both have a positive sign so it's going to be a to the one-third c to the one-fourth on top now b and d both contain a negative sign in front of the log so they're going to go on the bottom b to the two thirds and then d to the four fifths so you can leave your answer like this or you can convert it back into radical form so you can write it like this as well the cube root of a times the fourth root of c divided by the cube root of b squared times the fifth root of d to the fourth you