We want to solve the given exponential equations using the properties of exponents.
In these examples, we can write the left side and the right side of the equation
using the same base. Therefore, we can solve these equations without logarithms.
In general, if we have "a" raised to the power of "m" equal to "a" raised to the power of "n,"
then "m" is equal to "n." This means that if we have two exponential expressions that are
equal to each other and the bases are the same, then the exponents must be equal.
Looking at our first example, notice how our bases are not the same because this base is
three and this base is nine. But since nine is equal to three times three, or three squared,
we can rewrite this as three raised to the power of "2x + 5" equal to three squared raised to the
power of "3x - 7." Now notice we do have a common base of three, but on the right side,
we have three raised to the power of "3x - 7," or we have a power raised to a power, which means we
can multiply these exponents given by this rule here. So we can rewrite this as three raised
to the power of "2x + 5" equal to three raised to the power of "2 times the quantity 3x - 7."
In this form, now that the bases are the same and these are equal to each other,
"2x + 5" must equal "2 times the quantity 3x - 7." Now we can solve this equation for "x" to
determine our solution. So we'll start by clearing the parentheses here. So we'll have "2x + 5" equal
to "6x - 14." We need to get "x" on one side of the equation, so let's go ahead and subtract "2x"
on both sides of the equation. Notice how this would make the "x" terms on the right
side of the equation since this would be zero. So let's go ahead and add 14 here at the same time.
On the left side, we have 5 plus 14, which is 19. On the right side, we have "6x - 2x," which
is "4x," and this is zero as well. So now we'll divide both sides by four. We have "19/4" equal
to "x," which is our solution, or we can write "x" equal to "19/4" by switching the order.
Looking at our second example, we do have this fraction here to deal with, but we know four is
equal to two squared, so we could write this as "1 over 2 squared" raised to the power of "2x + 1."
And then on the right side, 64 can also be written as two raised to a power. 64 is equal
to eight times eight, and eight is equal to four times two, and four is equal to two times two. So
each eight contains three factors of two,
so 64 is equal to six factors of two, or two raised to the power of six.
Now, we could have actually left this as base four because 64 is also equal to four cubed,
but if we're ever in doubt, we can always break things down into their prime factors. Now again,
we still have this fraction to deal with. We want to have a common base of two on
both sides of the equation, so now what we can do is move "2 squared" up into the numerator,
and we rewrite this as "2 raised to the power of negative 2." Again, remember if we cross
a fraction bar, it's going to change the sign of the exponent, so we'd have "2 raised to the power
of negative 2" raised to the power of "2x + 1" equal to "2 raised to the power of six." Again,
on the left side, now we have powers raised to powers, so we're going to multiply these
exponents. So we'll have "2 raised to the power of negative 2 times the quantity 2x + 1" equal to "2
raised to the power of six." Now that we have these equal to each other and the bases are the
same, the exponents must be equal. So "negative 2 times the quantity 2x + 1" must equal six.
So now we'll clear the parentheses. We'll have "negative 4x - 2" equal to six. So
we'll add two to both sides of the equation. This will be zero, so we have "negative 4x"
equal to eight. Divide both sides by negative four, and we have "x" equal to negative two.
Okay, I hope you found this helpful.