Solving Logarithmic Equations/Operations in Math

Solving Logarithmic Equations/Operations in Math

WARNING

All words or numbers highlighted in red indicate that blogger has either (A) Resetted the features that shouldn't have been resetted or (B) deleted symbols and math equations (fraction lines, the square root symbol, ect.)

WHAT YOU MUST KNOW IN ORDER TO START SOLVING LOGARITHMS

• Exponents

• Square roots

• Multiplying

• Algebra

And, you must be comfortable with math.

What’s up everyone! I’m back today with another blog. This time I will be teaching you How to solve Logarithmic/Log Equations/Operations in math. Well, you might have heard of the “Log” equation, or seen it on a calculator. Let’s say log(8 = 0.903089987. Okay, well, we’re not going to go that deep into log(8 or log(2. Well, let’s start off with a question before we get started. (Oh, by the way, I'm very sorry for when I pasted this to blogger, it resetted literally everything, so it might be a little hard to read. And all the subscript in the equations are resetted.)

What is a Log Equation? When your using a log equation, it’s like exponents and algebra. When your using log, it’s like: What power do I need to raise to to get a number? Or, you could say, y to the x power equals z. I’ll draw them out.



The bottom picture is basically saying "what number to the power of 2 can I have to get 16? Or the bottom picture could be saying 2 to the x power equals 16." Either way is fine. Or, the top picture is saying that y to the x power equals z. (By the way the z on the top picture looks like a 2.) In a log, there is a base. The base is that small number written at the bottom. For example in the bottom drawing I made, 2 is the base because it is small and written at the bottom.

Here is an example: log₃ ⁸¹ = x. So, this is basically saying 3 to the x equals 81. So what power do we have to raise 3 to, to get to 81? We know that 1 can’t work. 3² is 9. 3³ is 27. So let’s try 3⁴. This is basically 3x3x3x3 which is the same as 9x9 which is 81. So now you have it, log₃ ⁸¹ = 4.  

Now try this one: logx ⁸¹ = 4. What is x? Try to take a moment and solve this on your own… So, this is basically saying that x to the 4th power equals 81. Write it out like this: X⁴ = 81. To do that, we need to take the 4th root, I believe, from each side. It’s the 4th root because it is X to the power of 4. So, it would be (X⁴)14 = (81)14. So, I would write it out like this: X = 4*81. Notice how that’s different from the division sign, so we’re not dividing. So the fourth root of 81 is 3. X = 3. *Excuse me, blogger deleted the square root sign there. So just imagine a square root sign between 4 and 81.

Let’s try an easier one: log₅ ˣ = 4. This is basically the same thing 5 to the 4th power equals x. Which means, 5x5x5x5 = x. 5x5 is 25, and 25x25= 625. So, x = 625.

Let’s try a harder one: log₃₂ ˣ = 45. This is basically the same thing as 32 to the power of 45 = x. So first, you want to find the 5th root of 32. Why is it the 5th? Because the denominator is 5. And then, raise it to the power of 4.Why 4? Because the numerator is 4. So what number raised to the 5th power equals 32? 2. 2x2x2x2x2 = 32. So the 5th root of 32 is 2. So now, we have to raise 2 to the power of 4. So 2⁴ = x. 2⁴ is 16. So x = 16.
Now see if you can figure out this one: log ˣ = 24. You might have noticed a problem: there is no base. In logarithms, if there is no base, it is assumed to be base 10. Why? Because on calculators, when you write a log equation, the calculator itself will assume it’s a base 10. Let’s say I typed in “log(100” on the calculator. I didn’t type in a base, so the calculator will assume it’s base 10. So when I see what the answer is to the problem, the calculator will say “2”. Because 10 to the 2nd is 100. So if you were to go back to the problem given, you would write it out like: log₁₀ ˣ = 32. Okay, the answer would be a ginormous number. So i’ll leave the answer as x = 10³².

I’ll give you a challenging one now: log (log x) = 3. Remember, if there’s no base, then the base is 10. So log₁₀ (log x) = 3. Now in this one, there’s 2 logs. For now don’t worry about the second log. This is basically the same thing as 10³ = (the numbers in the parentheses). So, 10³ = log x. 10 to the 3rd is 1000. So 1000 = log x. Hey, this second log doesn’t have a base. So what do we put as base? 10. So 1000 = log₁₀ x. In this case where I first found confusing, you put it in base 10 of 1000, if I’m saying it right. So, 10¹⁰⁰⁰ = x.

Now try this one: log₆₄ ⁴ = x. So let’s think of this as 64 to the x power = 4. Well, this might be a tricky one for beginners. So what power do you have to raise 64 to, to get to 4? 64 to the 2nd is 4096, which is way more than what we were looking for. In fact way more that we might have to go below 1. So let’s think about log₄ ⁶⁴ = x. What power do you have to raise 4 to get 64? 3. 4x4x4 = 64. So when we’re doing problems like log₆₄ ⁴ = x, all we do is flip the reciprocal of 3. So, 64 to the *1/3 power equals 4.
                     *I meant to put a fraction line dividing the 1 and 3, with 1 as numerator and 3 as denominator. But blogger has deleted it. So I will put a slash between the 1 and 3. 

Summary:

• A logarithm is an equation with algebra and exponents. To solve one, think of y to the x power equals z, shown on the picture from the top. Or, you could think of, what power do I need to raise y to, to get z?

• Every logarithm has a base, like base y of z equals x.

I think we have a bit of a warm up here. Now, I will give you a few logarithm problems, and try solving them on your own. The answer key is at the end of this blog. We will start out with some basic ones, and then progress on to harder ones.




Log₃ ˣ = 4




Log ₓ ⁶⁴ = 3




Log ˣ = 2




Log₃₂ ² = x




I think I won’t make this blog too long. I’ll end it here. If you have any questions, please write it in the comments section. The answer key is below. I hope you enjoy learning logarithms. BYEE!

Answer Key:

81

4

100

*1/5
*There was a fraction line dividing the 1 and 5, with 1 as numerator and 5 as denominator. But blogger has deleted the fraction line, so I will put a slash between the 1 and 5.