What is the Correct Notation for Logarithmic Equations?

[ThatDude]

Homework Statement

(2^(x+1)) / (5^x)) = 3


2.The attempt at a solution

(2^(x+1)) / (5^x)) = 3

((log 2 (x+1)) / (x log 5)) = log 3

(x log 2 + log 2) / (x log 5) = log 3

( log (2^ (x+1))) / (log (5^x)) = log 3

log (base: 5^x) (number: 2 ^ (x+1)) = log 3

(5^x) ^ (log 3) = 2 ^ (x+1)

(x log 5) ( log 3) = x log 2 + log 2

x ~ 9.09

I know that this is not the right answer but I don't understand where I went wrong.

 

[tiny-tim]

Hi ThatDude!

(try using the X² button just above the Reply box )

(2^(x+1)) / (5^x)) = 3

((log 2 (x+1)) / (x log 5)) = log 3

((log 2 (x+1)) minus (x log 5)) = log 3

btw, it would have been simpler to start by expressing the LHS as a multiple of (2/5)^x ​

 

[ThatDude]

Hi ThatDude!

(try using the X² button just above the Reply box )


((log 2 (x+1)) minus (x log 5)) = log 3

btw, it would have been simpler to start by expressing the LHS as a multiple of (2/5)^x ​

Thank you for helping me out.

I thought that you could only subtract two logs when the number inside the log was being divided. In this case, isn't it the division of two logs, albeit with the same base?

Ex:
= log5 ^x/12

= log5^x - log5¹²

But from what I understand from your post, the following is also true:


log5 ^x/12 = log5^x / log5 ¹²

 

[tiny-tim]

I thought that you could only subtract two logs when the number inside the log was being divided.

it is

you confused yourself by leaving out a step …

2^x+1 / 5^x = 3

log (2^x+1 / 5^x) = log 3

log (2^x+1) - log (5^x) = log 3 ​

 

[SteamKing]

Thank you for helping me out.

I thought that you could only subtract two logs when the number inside the log was being divided. In this case, isn't it the division of two logs, albeit with the same base?

Ex:
= log5 ^x/12

= log5^x - log5¹²

But from what I understand from your post, the following is also true:


log5 ^x/12 = log5^x / log5 ¹²

You have some confusing expressions:

What does log5 ^x/12 mean?

Is it log (5^x/12) or log$_{5}(x/12)$?

 

[Ray Vickson]

Thank you for helping me out.

I thought that you could only subtract two logs when the number inside the log was being divided. In this case, isn't it the division of two logs, albeit with the same base?

Ex:
= log5 ^x/12

= log5^x - log5¹²

But from what I understand from your post, the following is also true:


log5 ^x/12 = log5^x / log5 ¹²

It is impossible to say whether you are correct, or not, because of the poor notation. If $\log 5^{x/12}$ means $\log_5 (x/12)$ then what you wrote is true. If it means $\log \left( 5^{x/12}\right)$ then what you wrote is false.

 

[ThatDude]

it is

you confused yourself by leaving out a step …

2^x+1 / 5^x = 3

log (2^x+1 / 5^x) = log 3

log (2^x+1) - log (5^x) = log 3 ​

Ah. I see. Thank you for your help.

Yes... Ray Vickson and SteamKing, it is log5(x/12) ; that is, the base is 5 and the argument is x divided by 12. I apologize for the poor notation.

 

[Ray Vickson]

Ah. I see. Thank you for your help.

Yes... Ray Vickson and SteamKing, it is log5(x/12) ; that is, the base is 5 and the argument is x divided by 12. I apologize for the poor notation.

In that case you should write something like log[5] x/12 or log_5 x/12 or log₅ x/12; just about everybody would "get it" if you wrote it in any of these three ways (although it would not hurt to also include a brief verbal description, saying what the '5' means). It would be even better to include parentheses, like this: log[5] (x/12) or log_5 (x/12) or log₅ (x/12).