What is a log?

Let’s first talk about radicals, like square roots, cube roots, etc. We’ll focus on square roots because that’s easiest understood.

  • We clearly understood what it means to square a number, just multiply the number by itself.

Example: 32 means 3 · 3 which equals 9
Looking at this example as an equation:

y = x2for x = 3, we would substitute 3 in for x and solve
y = 32
y = 9

This was easy; give me any value for x and finding y is straight forward.

  • Now let’s look at it from a different perspective. What if you know y but you are looking for x.

y = x2for 16 = 32, we would substitute 16 in for y
16 = x2
x =?We’re looking for a number (x), such that when you square it (multiply it by itself), the answer is 16. For this problem it’s easy because 16 is a perfect square. So, the answer is:
x = 4

But what if y is not a perfect square? Let’s say y = 17

y = x2
17 = x2

x =?This one is not so easy. We do know that the answer for x is somewhere between 4 and 5 because:

x (guesses) / y = x2 / we want an answer of 17 for y
4 / 16 / too low
5 / 25 / too high
4.5 / 20.25 / too high
4.1 / 16.81 / too low
4.3 / 18.49 / too high
4.2 / 17.64 / too high
4.15 / 17.2225 / too high
4.12 / 16.9744 / too low
4.121 / 16.98264 / better but still too low
4.122 / 16.99088 / we're almost there

42 = 16
52 = 26

We could guess and get closer to 17. Here we’ll use excel.

So, this is why radicals were born, the most popular being square root (). Square root is a mathematical function that answers the question what number times itself (the x value) is equal to the number I have (the y value).

For our problem:

If were looking for the cube root (what number times itself is equal to the given y), It would look like this:

Notice closely the placement of the numbers.

We can take the root of any number given. We just need to know the y value and the exponent in order to set up the equation. In general:

Radicals are just mathematic functions that, in a sense, undo the exponents.

Now let’s look at logs.

This time my equation looks like:The exponential…. remember?

“a” and “b” are just numbers associated with my model. Let’s make the equation even simpler. Let’s remove “a”.
Now, let’s put in a value for “b”.

Assume that b = 10 (I just made that up)

y = 10x
x (given) / Solve for y
0 / 1
1 / 10
0.5 / 3.162278
0.3 / 1.995262
0.32 / 2.089296

This represents my model for “something (??).” I can put values in for “x” all day long and find the “y’s”. I’ll use excel.

y= 10x
Solve for x / y(given)
x=? / 2
x=? / 4
x=? / 8
x=? / 10
x=? / 3.9

But if I’m given “y” values and have to find “x” values, it is much more difficult. Let’s say for a moment I don’t know how to use my calculator to solve this.

Looking at the first entry for “y” (2), the question is “what value can I put in for “x” which will equal 2?”

I know the answer is somewhere between 0 and 1.

y = 10x
x (guesses) / y / we want an answer of y = 2
1 / 10 / too high
0.5 / 3.162278 / too high
0.3 / 1.995262 / too low
0.35 / 2.238721 / too high
0.31 / 2.041738 / close but still too high

We’ll use excel for some guessing.

This is a lot of work and we’re still just guessing. This is why logs (logarithms) were created.

A logarithm is a mathematical function we use to find exponents.The log IS the exponent.

Back to the problem we were guessing on.

If I’m looking for an exponent, I can use the log function.
Rewrite the equation so it looks like this: read: “x equals the log of 2”.

If you enter “log (2)” in your calculator it will automatically determine the correct exponent to which you raise 10 that will equal 2.

I can check this answer by substituting into the original equation (don’t clear you calculator with the last answer; if you do you will get round off error).

You need to notice the setup between the two equations. Pay close attention to where the numbers go.

Now, you should be thinking, “What about the 10?”

Whenever the subscript for a log is blank, it is understood to be 10.

If it is ever something other than 10, you must so state.

Example:
Here is how you enter a different base for a logarithm in your calculator ( got this from this website :
This would be for a problem that looks like this: (the answer should be 6)

So, here’s how your work should look when you are using logs to find exponents.

Another great website from Kahn Academy: