Again a logarithmic inequality

[Saitama]

Homework Statement

i got stuck at the question below:-

Homework Equations

The Attempt at a Solution

I tried to solve it by simplifying it but i got stuck at:-

Please help.

 

[tiny-tim]

Hi Pranav-Arora!

Just simplify the bottom …

what is the difference between log₃(9 - 3^x) and log₃(1 - 3^x-2) ?

 

[Saitama]

Hi Pranav-Arora!

Just simplify the bottom …

what is the difference between log₃(9 - 3^x) and log₃(1 - 3^x-2) ?

Sorry! Didn't get you...

 

[SammyS]

One method for solving an inequality is to solve the associated equation; in this case that's

$$\frac{x-1}{\log_3(9-3^x)-3}=1$$.

Then the critical numbers are the solution set and any points of discontinuity.

Use test points (in the domain of the left hand side of the inequality) which either to the laeft or right of all the test points or between any pair of test point.

By the way, what is log₃(9(1-3^x-2)) ?

 

[Saitama]

One method for solving an inequality is to solve the associated equation; in this case that's

$$\frac{x-1}{\log_3(9-3^x)-3}=1$$.

Then the critical numbers are the solution set and any points of discontinuity.

Use test points (in the domain of the left hand side of the inequality) which either to the laeft or right of all the test points or between any pair of test point.

By the way, what is log₃(9(1-3^x-2)) ?

i think i forgot to mention, i need to find out the values of x.

 

[tiny-tim]

Sorry! Didn't get you...

Hi Pranav-Arora!

what is log₃(9 - 3^x) - log₃(1 - 3^x-2) ?

 

[Saitama]

Hi Pranav-Arora!

what is log₃(9 - 3^x) - log₃(1 - 3^x-2) ?

It would be

9 - 3^x
-------
1 - 3^x-2

But why i need to find this?

 

[tiny-tim]

What is

9 - 3^x
-------
1 - 3^x-2​

? ​

 

[SammyS]

Try graphing $$\frac{x-1}{\log_3(9-3^x)-3}$$ or $$\frac{x-1}{\log_3(9-3^x)-3}-1\,.$$

Remember that $$\log_3(a)=\frac{\ln(a)}{\ln(3)}$$

 

[Saitama]

Try graphing $$\frac{x-1}{\log_3(9-3^x)-3}$$ or $$\frac{x-1}{\log_3(9-3^x)-3}-1\,.$$

Remember that $$\log_3(a)=\frac{\ln(a)}{\ln(3)}$$

Whoops! i forgot it:-
$$\log_3(a)=\frac{\ln(a)}{\ln(3)}$$

I converted everything in log and then i was able to figure it out.

Thanks SammyS...:)