Really need help understanding this log equation

Question

Hello, Im trying to solve for x. Its been days now I try to understand but I cant seem to. I understand how to solve for x for my other problems but this one is just really tough for me. So it says to solve for x for : 

 

Ln(x)-Log(x) =1 

 

Where I get stuck is after finding the LCD of the fraction and simplyfying. After this I am clueless. 

 

Im really scared this one will be on test so Im doing my best to understand

Answer 1

Convert logx to  (ln x)/ln 10...using "Change of Base".

 

Then you can factor out ln x on left side, etc...

Answer 2

Kyle,

 

The difficulty with this question is that you see two logarithms with different bases. Remember that ln is log_e and log is log₁₀. What you must do is write one of these in terms of the base of the other logarithm. Since e is the base most used in advanced math (calculus, etc.), I will show you how to do this using ln only.

 

Remember the Change of Base formula:

 

log_bx = (log_ax)/(log_ab).

 

If you don't know already, the idea is to compute (or rewrite) log_bx in terms of log_a. Here, you would use b = 10 and a = e. Thus,

 

log x = (ln x)/(ln 10).

 

When you go back to the original equation and replace log(x) by this, you get the new equation.

 

ln x − (ln x)/(ln 10) = 1.

 

To solve for x, you must first gather all ln x terms together and factor it out from them:

 

(ln x)(1 − 1/ln 10) = 1.

 

You can next divide by the second factor on the left side of the equation:

 

ln x = 1/(1 − 1/ln 10).

 

Although this looks a little messy, you can simplify it by getting the denominator (under the 1) on the right rewritten with a common denominator.

 

1 − 1/ln 10 = (ln 10 − 1)/(ln 10).

 

You can then take its reciprocal by flipping the fraction upside down:

 

ln x = (ln 10)/(ln 10 − 1).

 

Now, to solve for x, you exponentiate both sides. That means, raise e to both sides and set the results equal to each other.

 

x = e^{ln x} = exp [(ln 10)/(ln 10 − 1)].

 

As for simplifying the answer, you can use a calculator and get x ≈ 1.7677. Due to the exponent being a fraction, there is no easy way to simplify the answer to be put in an exact form.