Solving for Logb(16) with Logb(2) = 0.4307

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Homework Help Overview

The discussion revolves around finding logb(16) given that logb(2) = 0.4307. Participants explore the relationship between logarithmic values and their bases, particularly focusing on the implications of the base and exponent in logarithmic equations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss whether they can eliminate the logarithm to solve for the unknown, with some questioning the validity of their approaches. There is an exploration of the relationship between the base and the exponent, particularly in the context of expressing 16 as a power of 2. Others express confusion about the necessity of finding the base b and suggest alternative methods to approach the problem.

Discussion Status

The discussion includes various attempts to solve the problem, with some participants providing hints and guidance without reaching a consensus. There is acknowledgment of correct calculations, but also a recognition of the need for clearer methods to derive the logarithmic values without trial and error.

Contextual Notes

Participants note that the problem does not explicitly require finding the base b, which leads to some confusion regarding the steps necessary to solve for logb(16). There is also a hint that the problem may involve using properties of logarithms, such as the product rule.

yoleven
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Homework Statement


If logb(2)=0.4307, find logb(16)


Homework Equations





The Attempt at a Solution


If the log has the same base can I eliminate it and solve the equation?
If b^.04307=2 then b^?=16
can I say 2/.04307=16/x
16*0.4307=2x
x=3.4456
Am I close or have I missed something obvious?
 
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yoleven said:
If logb(2)=0.4307, find logb(16)

can I say 2/.04307=16/x

NO!

Hint: 16 = 2 x 2 x 2 x 2. :smile:
 
tiny-tim said:
NO!

Hint: 16 = 2 x 2 x 2 x 2. :smile:

Okay, 2^4=16
If I have b^.4307=2 By trial and error, I came up with 5 for b. 5^.4307=2 or log5(.4307)=2
Specifically, what steps do I follow to discover what the base is without resorting to a trial and error method.
Thanks
 
Start with b^.4307=2, and raise both sides to a certain power, such that you will get 16 on the right.
 
yoleven said:
If logb(2)=0.4307, find logb(16)
yoleven said:
Okay, 2^4=16
If I have b^.4307=2 By trial and error, I came up with 5 for b. 5^.4307=2 or log5(.4307)=2
Specifically, what steps do I follow to discover what the base is without resorting to a trial and error method.

Hi yoleven! :smile:

You don't need to find b … the question doesn't ask you for b.

Hint: 16 = 2 x 2 x 2 x 2.

logb(pq) = logb(p) + logb(q) :smile:
 
Or, more simply for this problem log(ab)= b log(a).
 
HallsofIvy said:
Or, more simply for this problem log(ab)= b log(a).

oooh … that's far too advanced! :wink:
 
If logb(2)=.4307
logb(16)=1.7228
because if b^.4307=2, (b^.4307)^4=(2)^4
b^1.7228=16
Okay? Thanks again.
 
1.7228 is correct.
 

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