Logarithms – Fundamental Properties


The log of ‘a’ to the base ‘b’ is equal to ‘x’. How do we write this in the exponential form? We can write it as ‘b’ raised to ‘x’ is equal to ‘a’. Both the equations tell us the same thing. Now before we move forward there are a few fundamental properties we need to know. Let’s discuss the first one. What will log of 1 to the base ‘b’ be equal to? How do we find this answer? I encourage you to pause the video, and find the answer to this. Say it equals ‘x’. In the exponential form, it can be written as ‘b’ raised to ‘x’ is equal to 1. Remember, the exponential form is what most of us are comfortable with. This can help us find the value of ‘x’. For the answer to be 1, the power here has to be 0. Do you remember that property of exponents? Any non zero number raised to 0 is equal to 1. Hence, ‘x’ here is 0. The log of 1 to any base will always equal 0. Log of 1 to the base 12 is equal to 0. Log of 1 to the base 250 is also equal to 0. That was the first fundamental property we looked at. Log of 1 to any base will always equal 0. Now let’s move on to the second property. What will be the value of log of ‘b’ to the base ‘b’? Give it a shot. Assume that it equals ‘x’. So in the exponential form, it can be written as ‘b’ raised to ‘x’ is equal to ‘b’. What will be the value of ‘x’ here? Come on, it’s easy! ‘b’ to the power 1 will equal ‘b’. ‘x’ will be equal to 1. The log of ‘b’ to the base ‘b’ will equal 1. If the argument and the base are the same, then the logarithm will be 1. For instance, the log of 4 to the base 4 will be equal to 1! That was the second property. The third one is pretty cool. It talks about the log of ‘b’ raised to ‘n’ to the base ‘b’. Assuming this equals ‘x’ try writing this in the exponential form. It is written as ‘b’ raised to ‘x’ is equal to ‘b’ raised to ‘n’. It’s pretty clear. ‘x’ has to be equal to ‘n’. So the log of ‘b’ raised to ‘n’ to the base ‘b’ will equal ‘n’. An example would be log of 11 cube to the base 11 will equal 3. So these are the three fundamental properties of logarithms, and we don’t really need to remember these as they can be easily derived. Along with the properties, there are a few conditions we need to know about. Let’s see them on the right side of the board. The basic form of logarithm is written like this, where ‘a’ is the argument and ‘b’ is the base! Now for this to be valid, there are a few conditions about ‘a’ and ‘b’ that we need to know. Okay, first the argument has to be greater than zero. This value here has to be greater than zero. So log of 0 to the base ‘b’ is undefined, and the log of a negative number to the base ‘b’ is also incorrect. But don’t get confused here. The answer of the log can be 0 or negative. For example, as we saw here the log of 1 to the base ‘b’ will be 0. This is correct. And what will be log of 1 half to the base 2? It will equal ‘negative 1’ as ‘2 raised to negative 1’ is equal to ‘1 over two’ or 0.5. This is also correct. The answer can be 0 or negative. But the argument has to be greater than 0. What are the conditions for ‘b’? The base ‘b’ has to be greater than 0. And it cannot be equal to 1. Yes, the base ‘b’ cannot be equal to 1. So the log of ‘a’ to the negative number base is incorrect. Why is 1 not allowed as the base? I want you to tax your brain a bit and come up with an answer. Why can’t the base be equal to 1? Let’s say the base was 1 and we have a logarithmic expression like log of ‘a’ to the base 1 is equal to ‘x’. In the exponential form it can be written as 1 raised to ‘x’ is equal to ‘a’. What can be the value of ‘x’ that gives us ‘a’? If ‘a’ is 2, what will ‘x’ be? If ‘a’ is 5, what will ‘x’ be equal to? We can’t really come up with an answer because 1 raised to any number will result in a 1! Hence this will be undefined. So these were the three simple primary properties of logs, and these were the conditions. With this, we can move ahead and solve interesting examples.

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