What power do you raise 3 to to get 9

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In summary, Daniel is trying to figure out what power he needs to raise 16 to get 2. He uses his calculator to try different solutions, but all of them give him a number that is not 2. He eventually finds that 0.25 is the correct answer.
  • #1
m0286
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Hello again..
Alright I am now on the part of 12th grade calculus, dealing with logs and exponents and stuff. I understand if say log39 That its basically what power do you raise 3 to to get 9. Well a question i need to answer is log162. So what power to I raise 16 to to get 2? I used my calculator just trying to find a number that would work, and i got 0.25. How would I show work for that? all of the solutions they gave for different equations worked different like for the log39They showed 2 ways:
3^x=9
3^x=3^2
x=2
OR
log39=log3(3^2)
=2
None of these solutions work for 16 since 2 is 2^1 and 16^1 is 16 not 2... I am confused.. Is 0.25 even the right answer.
Any Help is appreciated greatly!
Thanks ya!
 
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  • #2
hint: 16=2^4
 
  • #3
Yes.Use this property of the logarithms wrt a change of basis.

[tex] \log_{16}2=\frac{\ln 2}{\ln 16}=\frac{\ln 2}{4\ln 2}=\frac{1}{4}=0.25 [/tex]

Daniel.
 
  • #4
Have you heard of a change of base? You calculator can only do log base 10. Here is the formula

[tex]\log_b a = \frac {\log_c a}{\log_c b}[/tex] where c can be anything. But why not set that to 10 so your calculator can crunch it.
 
  • #5
0.25 is the correct answer, because 16^(0.25) = 2.

The easiest way to compute logs like [itex]\log_b x[/itex] on a calculator is to compute

[tex]\log_b x = \frac{\log x}{\log b}[/tex]

where the logs on the right can be any base at all (10, e, whatever). In other words, to find [itex]\log_{16} 2[/itex] on your calculator, punch up (log 2 / log 16).

- Warren
 
  • #6
Thanks, but what does the In stand for...
 
  • #7
I don't think so.Any (scientifical) calculator should have natural logarithm.It's eseential.

Daniel.
 
  • #8
m0286 said:
Thanks, but what does the In stand for...

That's probably "ln", the "natural logarithm", and it means the logarithm to the base e=2.1718..., if you haven't met it in your class yet, it's probably not far off.
 
  • #9
Dexter's post using natural logs can just as well be done with logs of any base, including 10.

- Warren
 

1. What is the power that you raise 3 to in order to get 9?

The power that you raise 3 to in order to get 9 is 2. This can be expressed as 32 = 9.

2. How do you calculate the power of 3 that results in 9?

To calculate the power of 3 that results in 9, you can use the logarithm function. Specifically, the base 3 logarithm of 9 is equal to 2.

3. Can you explain the relationship between 3, the power, and 9 in this equation?

The relationship between 3, the power, and 9 in this equation is that 3 raised to the power of 2 (or 32) results in 9. In other words, 2 is the exponent or power that is used to raise 3 to get 9.

4. How can I generalize this concept to solve similar equations with different numbers?

To generalize this concept, you can use the formula ax = b, where a is the base, x is the power, and b is the result. In the case of our example, a = 3, x = 2, and b = 9. So, in general, to solve for x, you can use the formula x = logab.

5. What is the significance of this equation in mathematics?

This equation is significant in mathematics because it demonstrates the fundamental concept of exponentiation. It shows how a base number can be raised to a certain power to result in a specific number. This concept is used in many mathematical and scientific calculations and is essential in understanding more complex equations and functions.

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