Now, we've already seen these in practice, but there are some helpful facts
that we can use to solve logarithmic equations. The first is that if a to the x
equals a to the y, then we know that x must equal y. We saw this earlier in the
problem 3 to the x equals 27. We rewrote 27 as 3 cubed. And since we have 3 to
the x equal to 3 to the y, we can set this exponent equal to this one. The next
fact that can help us solve equations with logarithms is that if a to the x
equals b to the x, then we know that the base a must equal the base b. We see
this in a problem like this. X squared equals 5 squared. Well we know since
these are both squares then x must be equal to 5. In order for this amount on
the left, to be equal to this amount on the right. The last fact that we'll use
is that if log of base b of n. Equals log base b of n, then we know that m must
equal n. Since we're taking the log of base b, we know that these two numbers
must be the same in order for the logarithms of those numbers to both be the
same as well. We haven't seen an example like this yet, but we'll use this idea
in the upcoming problems.
