We want to solve the given exponential equations using logarithms. If an exponential equation has
only one exponential part, like these two examples, we want to isolate the exponential
part and then take the log of both sides of the equation to solve for the variable.
So, looking at our first equation, we have 9 raised to the power of x plus 34 equals 94. Again,
our first step is to isolate the exponential part, or isolate 9 raised to the power of x.
We want to undo this addition, so we'll subtract 34 on both sides of
the equation. This should be zero, so now we have 9 raised to the power of x equals 60.
Now that we have the exponential part isolated, we're going to take the common log or natural
log of both sides of the equation. We use common log or natural log because we'll have to use the
calculator to solve this for x. Let's go and take the natural log of both sides of the equation.
And now, on the left side of the equation, we can apply the power property of logarithms and move
this exponent to the front. So now we have x times the natural log of 9 equals the natural log of 60.
And now, to solve for x, we can undo this multiplication by dividing both
sides by the natural log of 9. On the left side, this simplifies to one,
so we have x equals this quotient here, which we'll have to get a decimal approximation for
on the calculator. So we'll have the natural log of 60 divided by the natural log of 9.
It's standard to round to four decimal places, so x is approximately 1.8634. So
if we substitute this back into the original equation, this means that 9 raised to the
power of 1.8634 plus 34 should equal or be approximately 94. Let's go and check.
Because we did round down, the value is slightly less than 94, but this does verify our answer.
Looking at our second example, we want to isolate e raised to the power of 2x. So for
the first step, we'll undo the subtraction by adding 9 to both sides. So we have 4 times e
raised to the power of 2x equals 56. We also want to undo this multiplication
here between 4 and e raised to the power of 2x. So now we'll divide both sides by
4. This simplifies to one, so we have e raised to the power of 2x equals 14.
And now we'll take the log of both sides, and because we have base e here,
it'll be much easier if we take the natural log of both sides. So we'll
have the natural log of e raised to the power of 2x equals the natural log of 14.
And now we'll apply the power property of logarithms here so we can move this exponent
of 2x to the front. So this is now going to be 2x times the natural log of e equals the natural
log of 14. The natural log of e is actually equal to 1. Remember, natural log is base e,
and e to the first power is equal to e, verifying that the natural log of e is equal to 1.
So we could just drop this natural log e, but just in case we didn't recognize this,
we can now solve for x by dividing both sides by 2. The natural log of e divided by 2 over
2 simplifies to one. The natural log of e over the natural log of e simplifies to one.
So x is equal to this quotient here, which we'll use a calculator to get
a decimal approximation for. So our numerator is the natural log of 14,
and we'll divide this by 2. So x is approximately 1.3195, which means
4 times e raised to the power of 2 times 1.3195 minus 9 should be approximately equal to 47.
Let's go ahead and check. So second natural log brings up e raised to the power of 2 times 1.3195
minus 9, and it is approximately 47. So this does verify our solution.
Okay, we'll take a look at one more example in the next video. I hope you found this helpful.