"Simply put, a logarithm is the power that a number needs to be raised to to get some other number."
"For the log function, the domain is going to be from zero to infinity."
"This is just one illustration. Anything that has to do with ratios is going to encounter logarithms."
"Logarithms transform powers into multiplication, roots into division, and more."
"Logarithms themselves are more important than ever in science, mathematics, and engineering."
"Exponents inside the logs turn into constant factors, bases turn into constant factors."
"Whenever you take the log of 1, no matter the base, it's always going to equal zero."
"Exponents become where I can just take the exponent out and bring it on to the front times the logarithm of the basic expression."
"The nth prime number is roughly a trillion times natural log of a trillion."
"A logarithmic function is the inverse of an exponential function."
"Log base three of three is equal to one."
"Let's remind ourselves what P actually is... log base 3x is equal to -2."
"In order to get rid of a log, we need to raise both sides to the power of the base."
"Solve the equation two log base five of x take away log base five of three x equals two."
"The derivative of natural log x is 1 over x."
"So ln 10 over 11 is equal to negative x ln e, but ln e is one."
"Every time we see a logarithmic equation, you are going to need an exponential to solve it."
"Our final answer is x equals the natural log of two."
"If I want to come out from exponential to log, it's going to be log base two of eight is equal to three."
"Any log, regardless of any base as long as it's log one, regardless of any base is zero."
"Log is square root on steroids; it will take a big number and turn it small very quickly."
"If you have the subtraction symbol between two separate logarithms, that will allow you to consolidate these logarithms into a single quotient."
"If I wish to evaluate log base 9 of 27, I can write it as log 27 / log 9."
"Log base 3 of 9 is 2 because 3^2 is 9."
"Log base 3 of 27 is three, and so log base 9 of 27 you can reduce it to 3 over 2."
"The number e is intimately linked with logarithms."
"When we have the addition of two logarithms, then now what we can do is just go ahead and use that as a product."
"The natural log of e to the X is just going to be X."
"Log base two of eight equals three."
"Log base five of 125 is equal to 3 because 5 to the power of 3 is 125."
"Log base 4 of 1 over 64 is equal to negative 3."
"Don't be afraid of these logarithms... log 2 is just a decimal, you could just plug that into your calculator, it's just a number."
"A logarithm is the way you deal with this thing right here."
"If we're talking about log base B of X equals Y, here's how you can think about this in English."
"The integral of one over x dx is the natural log of x plus C."
"When finding the domain of logarithmic functions, you need to set the inside part greater than zero."
"It's very critical that you understand how to work with logarithms and exponential functions because they're very widely used and important."
"The logarithm here is a way to induce diminishing returns."
"Log of 100 is equal to 10 times log of a."
"The only thing that works is when it's $$ n \log n - n + \frac{1}{2}\log n + \text{some constant} $$."
"Everything you ever wanted to know about logarithms but were afraid to ask, we're going to try to cover in this video."
"The graph of a logarithm looks like y equals log base a of our variable x."
"Log base 10 of 1000 is a fancy way for writing the number 3."
"If you have two things being multiplied, you can break up the logarithm into the logarithm of one thing plus the logarithm of the other thing."
"If you have something raised to a power, this power can come out front as a factor."
"Log of x is understood to mean log base 10 of x."
"If you have a logarithm log base a of x and then you put that inside an exponential, that's going to also equal x."
"That's the only thing you can do with logarithms, to expand them or combine them."
"Any number in front of a logarithm is automatically an exponent."
"Logarithms are just a different way to represent an exponent."
"Log base 7 of the cube root of 4 is log base 7 of 4 to the 1/3 power."
"Any exponent can be moved to the front of our logarithm and be multiplied."
"And then that can be written as two u plus the natural log of u minus one minus the natural log of u plus one."
"Log base 2 of 1/8 is asking you a question: it's saying what power needs to be applied to 2 in order to make 1/8."
"Two logarithms being added can create one logarithm that has a product in that argument."
"Subtraction creates a quotient inside one logarithm, not two."
"Okay, let's go ahead and solve this equation using logarithms."
"Exponential functions are one to one, therefore they're going to have this inverse we're going to call that inverse a logarithm."
"Next time, we're going to start talking about what a logarithm is... and we'll talk about how they're the inverse of an exponential."
"If you're switching between logs and exponentials, base to the answer is equal to the bracket."
"Maximizing f(x) is the same as maximizing the log of f(x)."
"The answer to the logarithmic expression is always an exponent."
"When your unknown is in the exponent, you can get it out of the exponent by using the power rule of logarithms."
"Our true strain can be expressed as the natural logarithm of one plus our engineering strain."
"A to the log base a of X is simply X."
"To undo the log, you use its inverse function, the exponent to the same base."
"If the numerator is the derivatives of the denominator, then it will go back to an Ln of blah."
"The number of primes less than n converges to n over log n as n increases."
"Our temperature varies according to a constant multiplied by the natural log of the radius plus another constant of integration."
"The integral of log x is x log x minus x."
"We're generally going to be interested in the logarithm of n factorial."
"Undoing an exponential power is to be done using a logarithmic function."
"When you have something confusing with exponentials, take the logs and see what we get."
"E counters LN because E is the base of the natural logarithm."
"Upon integration, we know that the integral of Ln X or DX over X is Ln of X."
"When you do this integral right here, you're going to get natural log of P, which is base e, and when we solve that for P, you kind of get something on e, which is really kind of cool."
"Natural log of 6 is going to equal 10K, so K equals natural log of 6 over 10, and that's about 0.179."
"Logarithms are really simple, but we need to use them in this course."
"The domain of this is going to become the range of our log because they are inverses."
"When you want to solve an exponential, you use logarithms."
"If you recognize that the numerator is the derivative of the denominator, you can write that as the natural log of the absolute value of u plus c."
"The indefinite integral of 1/u du is going to be equal to the natural log of the absolute value of u plus c."
"You cannot take the natural log of a negative number."
"The indefinite integral of 1/x dx is equal to the natural log of the absolute value of x plus c."
"The prime counting function is approximately equal to X divided by the natural log of X."
"The indefinite integral of du/u is equal to the natural log of the absolute value of our denominator, our u plus c."
"Log n factorial is big theta n log n."
"The log base 2 of 1/16 is negative 4."
"Logarithms are the inverse of exponential functions with the same base."
"The most fundamental thing about logarithms in general is understanding that they're inverses of exponentials with the same base."
"The range of the logarithm function is all real numbers."
"Log of N factorial is big theta N log N."
"When you take the logarithm of a quotient, you can subtract the two individual logarithms."
"When you take the logarithm of some quantity raised to a power, then you can literally bring that power out so that it forms a product."
"If you have a product, you can break the logarithm into the sum of the two various logs."