Index Laws: Comparing 0.5^x & 2^x

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In summary, the task is to compare the graphs of 0.5^x and 2^x using index laws. The regular index laws may not directly apply, but it can be noted that 0.5^x can be written as 2^-x. This means that the two functions are reflections of each other in the Y-axis. This is a valid analysis according to the conversation.
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resurgance2001
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Homework Statement



we are asked to compare the graphs of 0.5^x with 2^x using index laws

Homework Equations



I am not sure how the regular index laws apply

The Attempt at a Solution



All I can say at the moment is the the graph of 2^x is a reflection of 0.5^x in the Y axis. I don't know how exactly to compare or justify this using index laws
 
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By "index laws", do you mean the rules for manipulation of quantities involving bases and exponents?

It may help to note that ##0.5 = \frac{1}{2} = 2^{-1}##.
 
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So then 0.5^x becomes 2^-x? So the two functions are effectively 2^x and 2^-x and because of this they are reflections of each other in the Y - axis.

I don't know what else the teacher might be expecting but I suspect that the answer the teacher is looking for. What do you reckon? The question was a bit vague.

Thanks
 
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  • #4
resurgance2001 said:
What do you reckon?
I think you are correct.
 
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  • #5
resurgance2001 said:
So then 0.5^x becomes 2^-x? So the two functions are effectively 2^x and 2^-x and because of this they are reflections of each other in the Y - axis.

Thanks
Yes. That's definitely a valid analysis.
 
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  • #6
SammyS said:
Yes. That's definitely a valid analysis.
Thank you confirming that.
 

1. What are index laws?

Index laws, also known as exponent laws, are mathematical rules used to simplify and manipulate expressions with exponents. They help us solve equations involving exponents and make calculations easier.

2. What is the difference between 0.5^x and 2^x?

The main difference between 0.5^x and 2^x is the base number. In 0.5^x, the base is 0.5, while in 2^x, the base is 2. This means that 0.5^x will always result in a number less than 1, while 2^x will always result in a number greater than 1.

3. How do you compare 0.5^x and 2^x?

In order to compare 0.5^x and 2^x, we can use the index laws. We know that 0.5^x can be written as (1/2)^x, and 2^x can be written as (2^1)^x. Using the index law (ab)^x = a^x * b^x, we can see that 0.5^x = (1^x/2^x) = 1/2^x. This means that 0.5^x is always smaller than 2^x, as the exponent of 2 is always greater than the exponent of 1/2.

4. What is the general rule for comparing exponents with different bases?

The general rule for comparing exponents with different bases is that when the bases are the same, the exponent with the larger base will result in a larger number. However, when the bases are different, we can use the index law a^x = (a^y)^z, where y and z are integers, to rewrite the expression with the same base. Then, we can compare the exponents to determine which expression is larger.

5. How do index laws apply to real-life situations?

Index laws are used in various fields of science, such as physics, chemistry, and biology, to make calculations and solve equations involving exponents. They are also used in finance to calculate compound interest and in computer science to represent and manipulate data. In everyday life, index laws can help us understand and analyze trends and patterns in data, such as population growth or stock market trends.

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