MHA1: Logarithms

Welcome to my very first Math Help Article, or simply MHA. Just to share something about my life, I and my classmates are entering the final stage of high school this coming school year; hence, most of us, if not all, are preparing for the college entrance exams with the best of our abilities. The graduates told us that the exams are really hard, and as I examined a review book, I must admit that the exams aren't so easy. It was then that I realized that, as I watched some of my friends struggle with their studies, it's about time that I share of what I've learned during my time as a Yahoo! Answers Top Contributor at the Science and Mathematics section.

As what you've read before, I am still a student, therefore, well, I'm not an official teacher. But please allow me to share the little information I have in my mind to those people who are having a hard time in Mathematics (I'll also make some Chemistry and Physics help articles in the future), because I'm also going through all of these; the confusion, the feeling of being light-headed, all the problems of a normal student, I also experience. The most convenient, yet the best way of sharing what I know to my classmates, and to the other people out there I can think of, is through this humble blog. Well, let's set this sentiment aside and let's proceed to studying about logarithms.

For those who haven't heard the word "logarithm," here's it's definition, taken from Wikipedia:

"In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce that number."

We can explain the definition via this statement: If log_bx = y, then bʸ= x

Where b is the base, y is the exponent, and x the number produced when b is raised to the y power. Let's have an example to explain things better.

Example 1: What is log 100?

Now, always remember this. If there is no subscript in the logarithmic notation (like example 1), the base is automatically 10. Some like to put a subscript 10 to be really sure that the base is 10, but it's good for you to know that in logarithms, the default base is 10.

Now, we recall this statement: If log_bx = y, then bʸ = x. We know that b = 10, and that x = 100. How can we find y? Well, we also know that bʸ should be equal to x, so 10ʸ should be equal to 100. So 10 should be raised to what number to have 100? Two! So y = 2.

Example 2: What is log 1000?

We use the same approach. We know that b = 10, and that x = 1000. Since bʸ should be equal to x, we also know that 10ʸ = 1000. So 10 should be raised to what number to have 1000? Well, 3, right? So y = 3.

Example 3: What is log₄64?

Now, the base isn't 10, but the same approach still works. We know that b is now 4, and that x = 64. Recalling the statement, we say that 4ʸ = 64. It isn't hard to figure out that 4 must be raised to the 3rd power to have 64. So, we conclude that y = 3.

Example 4: What is log₄2?

We know that b = 4, and that x = 2. 4 must be raised to what number to have 2? Well, I'm pretty sure everyone knows that √4 = 2. Also, if you still remember your basic algebra, we can express √4 as 4^(½).
So we can say that y = 1/2.

Example 5: What is log₆64z?

b = 6 and x = 64z. We know that bʸ = x. 6 must be raised to what number to have 64z? Well, we don't know. 64z isn't an actual number. So the answer is just log₆64z, although we shall be able to express this in many forms (which I'll discuss later). You'll also see stuff like this in logarithmic equations (which will be also discussed later).
 
Here are some examples to keep you going.
1. What is log₅625? The answer is 4.
2. What is log₂512? The answer is 9.
3. What is log₆₄4? The answer is 1/3.
4. What is log₂8²? The answer is 6.