In this video, we introduce logarithmic functions by reflecting the curves of exponential functions in the line y equals x, describe and illustrate very elegant logarithm laws, related to exponential laws that were discussed in an earlier video, and see that logarithms are a useful device for decreasing the complexity of arithmetic. You might recall, to invert a function, we reflect its graph in the line y equals x, which interchanges the roles of x, the input, and y, the output. Let's try this with y equals e^x, where e is Euler's number. Flipping this transparency over, interchanging the horizontal and vertical axes, has the effect of reflecting in the line y equals x and produces the graph of the inverse function. The curve we get corresponds to a function we call the natural logarithm, often written as y equals ln of x. ln is an abbreviation, though the origin of this is historically obscure, possibly related to the French word naturel for natural, used as an adjective coming after the noun. The exponential function using e as the base is often also called the natural exponential function. The rule for the natural logarithm may also be written with this log notation, an instance of something more general that we come to shortly. Observe that the domain of y equals e^x is the whole real line. Even though the y values quickly disappear into the distance in this diagram, e^x is defined for all real numbers x. The range is the set of positive reals, only taking values greater than zero. When we reflect in the line y equals x, to get the inverse function, the roles of horizontal and vertical are interchanged, as are the roles of input and output and domain and range. So the domain, for the natural logarithm, becomes the range of the exponential function, which is R plus, and the range for the natural logarithm becomes the domain of the exponential function, which is all of R. Notice that we have this restriction on the natural logarithm, that you only input positive real numbers. In the real number system, it doesn't make sense to take the log of a negative number. That was a lot to absorb, so recapping, the domain and range of y equals e^x is the whole real line and the set of positive reals, respectively. And you can denote the positive reals using interval notation if you like. Inverting functions interchanges domain and range. Therefore, the domain and range of y equals ln of x are the set of positive reals and the set of all reals, respectively. Because they're inverses of each other, these functions undo each other in the following two ways. Firstly, the natural logarithm of e^x is just x for all real numbers x. Secondly, if we raise e to the power of log x, then we get x back again, provided x is positive. Now, what about other positive real numbers a, used as a base for an exponential function? We can find a direct connection to the natural exponential function by exploiting one of the inverse function properties. For any positive a, we have a equals e raised to the power of ln of a. So a^x becomes e to the ln of a, all to the x, which by an exponential law becomes e to the product of ln of a with x. And we can rewrite this as e to the x times ln of a to avoid some brackets. And then the curve, y equals a^x, becomes y equals e to the x ln of a, and it must have the same shape as the curve y equals e^x, because when multiplying the input x by a constant, ln of a, so the effect is just a rescaling of the horizontal x-axis. Here's a general picture of the curve y equals a^x for the base a being positive and greater than one. We never consider a equals one as it's not interesting, 1^x equals one produces a constant function, and for a between zero and one, we have to reflect this shape in the y-axis, related to something called exponential decay. But we'll save that up for the next video. As before, we can reflect in the line y equals x, to get the inverse function, now called the logarithm to the base a, and we use this log notation. In particular, when a equals e, this reduces to the natural logarithm. I'll just warn you that abbreviated notation for logarithms is not universal. Some people write simply log when they mean log to the base 10, and others, including some calculators, write log for log to the base e. Logarithms satisfy important laws that follow from the exponential laws which we discussed in the last video. The first law says that the logarithm of a product is the sum of the logarithms. The second law says that the logarithm of a quotient is the difference of the logarithms. The third law says that the logarithm of a kth power is the kth multiple of the logarithm, in the sense that the exponent k falls down to the front. These laws also have succinct forms using the notation for the natural logarithm. These laws decrease the complexity of arithmetic. Logs turn products into sums, converting multiplication into addition, which is simpler, quotients into differences, converting division into subtraction, which is simpler, and powers into multiples, converting exponentiation into multiplication, another simplification further up in the hierarchy of arithmetic. Here's some practice. Simplify log to the base 10 of the square root of a thousand. Now the square root of a thousand is not a nice simple number, so you might expect to need a calculator. But in fact, we can simplify this quickly using the laws that we've introduced. First observe that the square root of a thousand is a thousand raised to the power of one-half, and then bring the half out the front, because logarithms convert powers into multiples. But 1000 is 10 cubed, and log to the base 10 undoes the exponential function using base 10. So log of 10 cubed to the base 10 must be just three. So we get a half of three which is three on two. And if you used a calculator, it should show 1.5. Now, in the process of our development earlier, we used the fact that if a is positive and b is any real number, then a^b, is e to the b log a, which is quite amazing. It reduces questions about very general exponential expressions to natural logarithms and natural exponentials, in a certain sense reducing everything to the mathematics of Euler's number. There's another useful fact. Let a be a fixed positive base and x any positive real number. Then x equals a to the log of x to the base a, since the exponential undoes the logarithm. We can then take natural logs of both sides, but remember, logs turn powers into multiples. So, the right-hand side becomes log of x to the base a times ln of a, and then dividing both sides by ln of a, and rearranging, we get, finally, log of x to the base a is the fraction ln of x divided by ln of a. This means that we can work out logs to any base just using natural logs and fractions. We've managed a lot today. We first defined the natural logarithm as an inverse function by reflecting the curve y equals e^x in the line y equals x in the plane, so that we can do this also in general for y equals a^x for any positive real number a, which defines the logarithm to the base a, then discussed logarithm laws and how these decrease the complexity of arithmetic, and, finally, developed some formulae, which tell us how to calculate general logs and exponentials using natural logs and natural exponentials related to Euler's number. Please read the notes accompanying this video and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
