Finding the value of an exponential given that e^a=10 and e^b=4

In summary: Since ln(b)= log_a(b)/log_a(e), it follows that log_a(b)= ln(b) log_a(e). Hence, y= log_a(b)= log_a(b) log_a(e).
  • #1
phosgene
146
1

Homework Statement



Suppose that e^a=10 and e^b=4. Find the value of

[itex]loga(b)[/itex]

Homework Equations



[itex]ln(b)=loga(b)/loga(e)[/itex]

The Attempt at a Solution



So far I've only gotten to the above equation rearranged to:

[itex]ln(b) loga(e)=loga(b)/[/itex]. I'm unsure as to where I'm supposed to go from here...
 
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  • #2
A useful relation is loga b = logeb / logea.
How is logeb related to eb?
 
  • #3
Hmm, you can get e^b from it by going exp(exp(ln(b))), but I'm kinda stuck. I get up to this stage:

loga(b)=lnb/lna

exp(loga(b))=exp(lnb/lna)

I can re-arrange so that I get exp(loga(b))=(exp(lnb))1/lna)=b^1/lna, then I'm unsure where to go as I don't know what to do with the 1/lna.
 
  • #4
phosgene said:
Hmm, you can get e^b from it by going exp(exp(ln(b))), but I'm kinda stuck. I get up to this stage:

EDIT: That doesn't seem useful for the problem here. What you should use is b = ln eb.

loga(b)=lnb/lna

Find ln a just the way you found ln b and substitute it here.

exp(loga(b))=exp(lnb/lna)

I can re-arrange so that I get exp(loga(b))=(exp(lnb))1/lna)=b^1/lna, then I'm unsure where to go as I don't know what to do with the 1/lna.

You don't need to do this. :smile:
 
  • #5
Wait, I just realized that all I had to do was solve for a and b in the original equations and plug it in. Doh! Thanks for the help though :)
 
  • #6
phosgene said:
Wait, I just realized that all I had to do was solve for a and b in the original equations and plug it in. Doh! Thanks for the help though :)
And if you want to write it in terms of 4 and 10 then it's [itex] \frac{\ln \ln 4}{ \ln \ln 10}[/itex]
 
  • #7
phosgene said:

Homework Statement



Suppose that e^a=10 and e^b=4. Find the value of

loga(b)

Homework Equations



ln(b)=loga(b)/loga(e)

The Attempt at a Solution



So far I've only gotten to the above equation rearranged to:

ln(b) loga(e)=loga(b). I'm unsure as to where I'm supposed to go from here...
(Do NOT put the HTML "sub" tags inside "tex" tags, either do not use "tex" or use underscores, "_", in "tex".)
For any x, eln(x)= x so that [itex]a^x= e^{ln(a^x)}= e^{x ln(a)}[/itex]. Similarly, if [itex]y= log_a(b)[/itex], then [itex]b= a^y[/itex] so that [itex]b= e^{ln a^y}= e^{y ln a}[/itex]. Then [itex]ln(b)= y ln(a)[/itex] so that [itex]y= ln_a(b)= ln(b)/ln(a)[/itex].
 

FAQ: Finding the value of an exponential given that e^a=10 and e^b=4

What is the value of a and b?

The value of a is approximately 2.3026 and the value of b is approximately 1.3863.

How do you find the value of an exponential given e^a=10 and e^b=4?

To find the value of an exponential, you can use the property of logarithms that states logex = y if and only if ey = x. In this case, we can rearrange the equations to get a = ln(10) and b = ln(4). Then, we can use the property of logarithms again to get ea = 10 and eb = 4.

Can you explain the concept of e and how it relates to exponents?

The value of e is an important mathematical constant that is approximately equal to 2.71828. It is often referred to as the "natural base" and is used in many mathematical and scientific calculations. The relationship between e and exponents can be seen in the fact that the derivative of ex is also ex, making it a convenient base for exponential functions.

What is the significance of finding the value of an exponential given e^a=10 and e^b=4?

Finding the value of an exponential in this scenario allows us to solve for the unknown exponents a and b, which can be useful in various mathematical and scientific applications. It also highlights the relationship between the exponential function and the constant e.

Can this concept be applied to other exponential equations?

Yes, this concept of using logarithms to find the value of an exponential given certain conditions can be applied to other exponential equations as well. It is a fundamental concept in mathematics and is commonly used in fields such as finance, physics, and biology.

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