Solving Equation with Negative Exponent ( thank you)

Multiply through:10^{-2}<10^{-x}x>2 as required.In summary, when solving for x in the inequality (1/10)^(x - 1) < (1/10), there are two methods that can be used, but only one will yield the correct answer of x > 2. The first method involves multiplying exponents, while the second method involves taking logs. However, in the second method, the inequality sign must be reversed if the base is less than 1. When the base is equal to 1,
  • #1
theanimux
2
0

Homework Statement



Solve for x

(1/10)^(x - 1) < (1/10)2. My Dilemma

The problem is that using one method will yield one answer, but using another method will yield another. Why is that? Please help.

My issue is not regarding the right answer (the right answer is x > 2). It's regarding the method/mathematical operation. Thanks!3. Solution attempts

Method 1: Multiplying exponents (this method should be the correct one)
Step 1: (10^-1)^(x - 1) < 10^-1
Step 2: multiply exponents:: 10^(-x + 1) < 10^-1
Step 3: simplify:: -x + 1 < -1
Step 4: solve:: x > 2

Method 2: Please tell me what's wrong with this method
Step 1: [(1/10)^x] / (1/10) < (1/10)
Step 2: multiply both sides by (1/10):: (1/10)^x < (1/10)^2
Step 3: solve:: x < 2
 
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  • #2
theanimux said:
Method 2: Please tell me what's wrong with this method
Step 1: [(1/10)^x] / (1/10) < (1/10)
Step 2: multiply both sides by (1/10):: (1/10)^x < (1/10)^2
Step 3: solve:: x < 2

Step 3 is wrong. From the step 2, you actually conclude that x > 2. If the base is less than zero, the inequality sign "changes direction" (as it does when you multiply inequality with -1).
 
  • #3
Do you mean when the base < 1? I tested cases, and when the base < 1, I would need to reverse the sign.

And when the base > 1, I can just continue order of operations like any normal equation.But when base = 1, then it appears that there is no solution. Do you have an explanation for that as well?Thanks a lot!
 
  • #4
theanimux said:
Do you mean when the base < 1? I tested cases, and when the base < 1, I would need to reverse the sign.

And when the base > 1, I can just continue order of operations like any normal equation.


But when base = 1, then it appears that there is no solution. Do you have an explanation for that as well?


Thanks a lot!

You are trying to solve an equation like a^x<a^2. You are doing it by implicitly taking logs. So x*log(a)<2*log(a). Now you want to divide by log(a). If log(a) is positive (a>1) you get x<2. If log(a) is negative (0<a<1) you have to reverse the inequality, so you get x>2. If a=1 then the original inequality is 1^x<1^2. To think about that problem, you don't need logs or anything.
 
  • #5
.1^(x-1) < .1
log (.1)^(x-1) < log .1
(x-1)(log .1) < log .1

Expand and solve.
 
  • #6
Logs aren't necessary.

[tex]\left(\frac{1}{10}\right)^x<\left(\frac{1}{10}\right)^2[/tex]

[tex]\frac{1}{10^x}<\frac{1}{10^2}[/tex]

Multiply through:

[tex]10^2<10^x[/tex]

[tex]x>2[/tex] as required.

Or you can even use the property that [tex]\frac{1}{a^x}=a^{-x}[/tex]
 

1. How do you solve equations with negative exponents?

To solve an equation with negative exponents, you can use the rule that states: a-n = 1/an. This means that you can rewrite the equation with the negative exponent as its reciprocal with a positive exponent. For example, x-3 = 1/x3. Once you have rewritten the equation, you can solve it using standard algebraic techniques.

2. What should I do if I encounter a negative exponent in a problem?

If you encounter a negative exponent in a problem, you should apply the rule mentioned above: a-n = 1/an. This will allow you to rewrite the equation with a positive exponent and continue solving it using standard algebraic techniques.

3. Can negative exponents be fractions?

Yes, negative exponents can be fractions. For example, x-1/2 can be rewritten as 1/x1/2. This is equivalent to 1/√x, which is a valid mathematical expression.

4. What is the difference between a negative exponent and a negative number?

A negative exponent refers to the placement of the exponent in a mathematical expression, whereas a negative number refers to the value of a number. For example, x-2 has a negative exponent, but it can represent a positive value if x is a positive number. On the other hand, -2 is a negative number that represents a value that is less than zero.

5. Can negative exponents be fractions with negative denominators?

No, negative exponents should not have negative denominators. This would result in a negative number in the denominator, which is undefined in mathematics. If you encounter a negative exponent with a negative denominator, you should apply the rule: a-n = 1/an and rewrite the fraction with a positive denominator.

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