yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Solving exponential equations using exponent properties | High School Math | Khan Academy


3m read
·Nov 11, 2024

Let's get some practice solving some exponential equations, and we have one right over here. We have (26^{9x + 5} = 1).

So pause the video and see if you can tell me what (x) is going to be. Well, the key here is to realize that (26^0) is equal to 1. Anything to the 0th power is going to be equal to one. Zero to the zero power we can discuss some other time, but anything other than zero to the zero power is going to be one.

So we just have to say, well, (9x + 5) needs to be equal to zero. (9x + 5) needs to be equal to zero, and this is pretty straightforward to solve. Subtract five from both sides, and we get (9x = -5). Divide both sides by nine, and we are left with (x = -\frac{5}{9}).

Let's do another one of these, and let's make it a little bit more interesting. Let's say we have the exponential equation (2^{3x + 5} = 64^{x - 7}).

Once again, pause the video and see if you can tell me what (x) is going to be or what (x) needs to be to satisfy this exponential equation.

All right, so you might at first say, "Oh, maybe (3x + 5) needs to be equal to (x - 7)," but that wouldn't work because these are two different bases. You have (2^{3x + 5}) then you have (64^{x - 7}).

So the key here is to express both of these with the same base, and lucky for us, (64) is a power of two. (2^3) is eight, so it's going to be (2^3 \times 2^3); eight times eight is sixty-four, so it's (2^6) is equal to sixty-four.

You can verify that. Take six twos and multiply them together, you’re going to get (64). This is just a little bit easier for me; eight times eight, and this is the same thing as (2^6) power is (64).

And I knew it was to the sixth power because I just added the exponents because I had the same base.

All right, so I can rewrite (64). Let me rewrite the whole thing. So this is (2^{x + 5} = 2^6), and then that to the (x - 7) power.

And to simplify this a little bit, we just have to remind ourselves that if I raise something to one power and then I raise that to another power, this is the same thing as raising my base to the product of these powers (a^{b \cdot c}).

So this equation I can rewrite as (2^{3x + 5} = 2^{6 \cdot (x - 7)}). So it's going to be (6x - (6 \cdot 7) = 42).

I'll just write the whole thing in yellow: (6x - 42). I just multiplied the (6) times the entire expression (x - 7).

And so now it's interesting. I have (2^{3x + 5}) power has to be equal to (2^{6x - 42}) power, so these need to be the same exponent. So (3x + 5) needs to be equal to (6x - 42).

So there we go; it sets up a nice little linear equation for us. (3x + 5 = 6x - 42).

Let's see, we could get all of our — since, well, I'll put all my (x)'s on the right-hand side since I have more (x)'s on the right already. So let me subtract (3x) from both sides, and let me — I want to get rid of this (42) here, so let's add (42) to both sides.

And we are going to be left with (5 + 42 = 47) is equal to (3x). Now we just divide both sides by (3), and we are left with (x = \frac{47}{3}).

(x = \frac{47}{3}), and we are done.

More Articles

View All
How to Hang a Tightrope Wire | StarTalk
Everyone’s first question would be: how do you get a wire from one building to another? If the wire is strong enough to hold your weight—not that you’re heavy—but if it’s strong enough to hold your weight, you can’t. You’re not—you can’t just feed, okay. …
"He Saved My Life" American Soldier Returns to Help Iraqi Captain Fleeing ISIS | National Geographic
[Music] [Music] Ian yes for [Music] I’m very scared to lose my son, lose my daughter, lose my wife, thus all my [Music] life. The soldiers, like the captain, are the ones that kept us alive. My name is Chase Msab. I’m a veteran of the Iraq War. I did thre…
YouTube changed my life (Started exactly one year ago today)
So you usually want to make a video. I’ll plan it out a little bit ahead of time, and I’ll make it like a format of what I’m gonna say and in what order, so don’t miss any points. Put a video like this, I figured it’s probably just best I just make a spu…
Rescuing a 14 Ton Bread Truck | Ice Road Rescue
NARRATOR: In the south, a 14-ton bread truck is impaled on rocks. Thord and Andrzej were attempting to lift it clear until it threatened to crash back down with Thord underneath. [bleep] that bloody left bar right there. [tools clanging] You know, we have…
Warren Buffett, Chairman, Berkshire Hathaway Investment Group | Terry Leadership Speaker Series
Good morning. It certainly got quiet quickly. That surprised me. Can you hear me? Are you there? Back well for business school, you know, it doesn’t get much better than this. Having the world’s greatest investor come to our campus is quite a bore. Office…
A Submarine Assault | WW2 Hell Under the Sea
July 31st, 1944. With Commander Lawson Ramage fixated on another target in Japanese convoy MI-11, below deck, battle helmsman Chet Stanton has made the decision to evade an escort that threatens to ram the American submarine. The crew of USS Parche wait t…