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Saitama
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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 said:Hi Pranav-Arora!
Just simplify the bottom …
what is the difference between log3(9 - 3x) and log3(1 - 3x-2) ?
SammyS said:One method for solving an inequality is to solve the associated equation; in this case that's
[tex]\frac{x-1}{\log_3(9-3^x)-3}=1[/tex].
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(1-3x-2)) ?
Pranav-Arora said:Sorry! Didn't get you...
tiny-tim said:Hi Pranav-Arora!
what is log3(9 - 3x) - log3(1 - 3x-2) ?
SammyS said:Try graphing [tex]\frac{x-1}{\log_3(9-3^x)-3}[/tex] or [tex]\frac{x-1}{\log_3(9-3^x)-3}-1\,.[/tex]
Remember that [tex]\log_3(a)=\frac{\ln(a)}{\ln(3)}[/tex]
A logarithmic inequality is an inequality that contains one or more logarithmic expressions. These expressions involve a logarithm, which is the inverse of an exponential function. In other words, a logarithmic inequality is an inequality that involves logarithms.
To solve a logarithmic inequality, you must first isolate the logarithmic expression on one side of the inequality. Then, you can rewrite the inequality in exponential form and use the properties of logarithms to simplify it. Finally, you can solve for the variable by using algebraic techniques.
The key properties of logarithms are the product rule, quotient rule, power rule, and the change of base rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. The power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. The change of base rule allows us to rewrite a logarithm in terms of any other base.
Logarithmic inequalities are important because they can model many real-world situations, such as population growth, radioactive decay, and sound intensity. They also have applications in mathematics, physics, and engineering. In addition, solving logarithmic inequalities can help improve critical thinking and problem-solving skills.
Yes, there are a few common mistakes that people make when solving logarithmic inequalities. These include forgetting to check for extraneous solutions, using the wrong base when applying the change of base rule, and incorrectly applying the properties of logarithms. It is important to carefully follow the steps and double-check your work to avoid these mistakes.