Authentic voices. Remarkable stories. AOL On Originals showcase the passions that make the world a more interesting place.
Go behind the scenes with some of the biggest digital celebrities to see what life is like when the blogging and tweeting stops.
The story of punk rock singer Laura Jane Grace of Against Me! who came out as a woman in 2012, and other members of the trans community whose experiences are woefully underrepresented and misunderstood in the media.
Documentary shorts conceived of and directed by famous actors. Jeff Garlin, Katie Holmes, Alia Shawkat, Judy Greer, and James Purefoy
Park Bench is a new kind of "talking show" straight from the mind of born and bred New Yorker and host, Steve Buscemi.
Digital influencer Justine Ezarik (iJustine) is back. After covering the world of wearable tech last season, iJustine is expanding her coverage this year by profiling the hottest tech trends across the country.
A 12 episode documentary series following 5 startup companies competing in the 2013 San Francisco TechCrunch Disrupt Startup Battlefield as they fine tune their products and eventually present in front of a panel of judges in hopes of winning $50,000 in funding.
Enter the graceful but competitive world of ballet through the eyes of executive producer, Sarah Jessica Parker. This behind-the-scenes docudrama reveals what it takes to perform on the ultimate stage, the New York City Ballet. Catch NYCB on stage at Lincoln Center.
Nicole Richie brings her unfiltered sense of humor and unique perspective to life in a new series based on her irreverent twitter feed. The show follows the outspoken celebrity as she shares her perspective on style, parenting, relationships and her journey to adulthood.
Explore what it means to be human as we rush head first into the future through the eyes, creativity, and mind of Tiffany Shlain, acclaimed filmmaker and speaker, founder of The Webby Awards, mother, constant pusher of boundaries and one of Newsweek’s “women shaping the 21st Century.”
Gwyneth Paltrow and Tracy Anderson spend time with women who've overcome hardship, injury, and setbacks to triumph in the face of adversity.
Hank Azaria’s touching, humorous, and often enlightening journey from a man who is not even sure he wants to have kids, to a father going through the joys, trials and tribulations of being a dad.
ACTING DISRUPTIVE takes viewers inside the businesses and passion projects of Hollywood’s top celebrities.
Hello! Let’s do some work on logarithm properties. So let’s just review real quickly what a logarithm even is. If I write log base x of A is equal to N, what does this mean? Well, this just means that x to the N equals A. I think we already know that, we’d learned that in the logarithm video. It’s very important to realize that when you evaluate a logarithm expression like log base x of A, the answer or when you evaluate what you get is an exponent. This N is really just an exponent; this is equal to this thing. You could have written just like this because this N is equal to this. You could just write x to the log base x of A is equal to A. All I did is I took this N and I replaced with this term. And I wanted to write it this way because I want you to really get an intuitive understanding of the notion that a logarithm when you evaluate it, it really is an exponent. And we’re going to take that notion and that’s where really all of the logarithm properties come from.
What I actually want to do is I want to stumble upon the logarithm properties by playing around it then later on I’ll summarize them and clean them all up. But I want to just show maybe how people originally discovered this stuff. So let’s say that x to the L is equal to A. Well if we write that as a logarithm, that’s same relationship as a logarithm, we could write that log base x of A is equal to L. If I were to say that x to the M is equal to B and the same thing, I just switched letters but that just means that log base x of B is equal to M. I just did the same thing that I did in this line, I just switched letters. So let’s just keep going and see what happens.
Let’s say I have x to the N is equal to A x B and that’s just like saying that log base x is equal to A x B. So what can we do with all of this? Well let’s start with this right here, x to the N is equal to A x B. So how can we rewrite this? Well A is this and B is this, so lt’s rewrite that. So we know that x to the N is equal to A, A is this, x to the L. And what’s B, times B, well B is x to the N. What x to the L times to the L? Well we know from the exponents when you multiply two expressions that have the same base and different exponents you just add the exponents. So this is equal to, when you have the same base and you’re multiplying, you can just add the exponents, that equals x to the L+M. So x to the N is equal to x to the L+M. Well we have the same base these exponents must equal each other. So we know that N is equal to L+M, and what does that do for us? What’s another way of writing N? So we said x tot eh N is equal to A x B that means that log base x of A x B is equal to N.
Anyway, so what’s N? What’s another way of writing N? Well another way of writing N is right here, log base x of A x B. Now we know that if we just substitute N for that we get log base x of A x B and what is that equal? That equals L, and another way to write L is right up here. It equals log base x of A+M, and what’s M? M is right here, log base x of B. And there we have our first logarithm property. The log base x of A x B well that just equals the log of base x of A plus the log base x of B. And this is hopefully, this proves that to you and if you want the intuition of why this works out it falls from the fact that logarithms are nothing but exponents.
So with that, I’ll leave you with this video and in the next video I will prove another logarithm property. I’ll see you soon.