Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator

Find the exact solution of the exponential equation in terms...

Key Concepts

Logarithmic Properties

Key properties of logarithms, such as the power rule (which states that log(a^b) = b · log(a)), are essential for manipulating and simplifying exponential equations. These properties help in expressing equations in a form that is more easily solvable.

Exact vs. Approximate Solutions

Exact solutions are expressed in a closed form, often in terms of logarithms or other constants, and provide a precise answer. Approximate solutions, on the other hand, involve numerical approximation methods—typically via calculators—to give a decimal value, which is especially useful for practical applications.

Logarithms

Logarithms are the inverses of exponential functions. They are crucial in solving exponential equations because applying a logarithm to both sides of an equation can bring the exponent down as a multiplier, simplifying the process of isolating the variable.

Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. Solving these equations often requires transforming the equation using logarithms to linearize the exponent, thereby making it easier to isolate the variable.

Transcript

00:02 All right, we have an exponential function here that we're trying to solve.

00:06 And this one's a little tricky because the two bases we have are different.

00:10 We have a base of two on the left and a base of three on the right.

00:13 Typically, we would get rid of a base with the matching log.

00:17 Here, we can't quite do that because we can't match both of these bases.

00:22 So what we're going to do is we are just going to use our natural log on both sides.

00:28 And this helps us out because our properties of logs say that we can take the x, on the inside out in front and turn it into a multiplication problem.

00:36 So it's going to end up being 3x plus 1 times the natural log of 2.

00:43 And that's going to equal x minus 2 times the natural log of 3.

00:50 And now we want to go ahead and get all of our xes on one side, all of the logs on the other.

00:54 So i'm going to divide both sides by natural log of 2 and then also divide both sides by x minus 2.

01:12 You know what? i don't know if that division will help.

01:14 I'm going to leave the x minus 2.

01:15 To let's go ahead and just leave things for now.

01:23 Thinking about this, because there are xes on both sides, that division is going to end us up with a fraction, and that's not going to be super helpful to us, but what we can do is we can go ahead and distribute both of these logs.

01:36 They are just numbers, really ugly numbers, but they're just numbers.

01:40 So if we distribute them, we would end up with three times a natural log of 2 times x, plus the natural log of 2 equals the natural log of 2, equals the number.

01:49 Natural log of 3 times x minus 2 natural log of 3.

01:52 And this looks really ugly, but remember these are just numbers.

01:56 And so what we can do is just our regular solving skills, where we get any constant that does not have an x to the same side, and then any variable that does have an x will go to the same side.

02:11 So we end up with 3 natural log of 2 times x minus natural log of 3 times x, equal to negative 2 natural log of 3 minus natural log of 2.

02:23 Again, it looks very ugly.

02:26 But how this subtraction works on the left is really, if we knew what these two numbers were, we could subtract them.

02:33 If it was like 5 minus 7, we would do 5 minus 7 and then put that in front of the x.

02:39 But we have these really ugly numbers that we don't want to deal with yet.

02:43 So we're just going to kind of pretend like we're doing that math.

02:45 We're going to take the first one and subtract the second one.

02:48 It would be another big ugly number, but we know it would just result in something in front of the x.

02:55 And now we can get rid of that coefficient that's multiplying in front of the x by dividing over here.

03:09 So we end up with x being equal to this phrase.

03:24 And this is something that we can actually combine using our log properties back together.

03:30 We can combine this back together into more of a singular log.

03:35 Again, anything multiplying out front is going to become an exponent on the inside...

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