Changing the bases of logs question

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In summary, changing the bases of logs refers to rewriting a logarithmic expression in a different base, often for simplification or solving equations. This can be done using the logarithm change of base formula, where any base can be used. It is important to keep in mind that the base cannot be negative, the argument must be positive, and to check for extraneous solutions when solving equations.
  • #1
lionely
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Homework Statement


If x= log2aa
y= log3a2a
z= log4a3a

prove that xyz +1 = 2yz

Homework Equations



loga C = log10C/log10a


The Attempt at a Solution



log= log10

x= log2aa = log a/ log2a


y= log3a2a = log 2a/log 3a


z= log4a3a = log 3a/log 4a


xyz + 1 = (log a/log2a)x (log 2a/log 3a) x (log 3a/log4a) + log 10


= log a/ log 4a + log 10

Is this correct so far?
 
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  • #2
lionely said:

Homework Statement


If x= log2aa
y= log3a2a
z= log4a3a

prove that xyz +1 = 2yz

Homework Equations



loga C = log10C/log10a


The Attempt at a Solution



log= log10

x= log2aa = log a/ log2a


y= log3a2a = log 2a/log 3a


z= log4a3a = log 3a/log 4a


xyz + 1 = (log a/log2a)x (log 2a/log 3a) x (log 3a/log4a) + log 10


= log a/ log 4a + log 10

Is this correct so far?

Everything you've written there is correct. But instead of writing log 10 in the last step, I would suggest you write the "1" as [itex]\frac{\log 4a}{\log 4a}[/itex] and combine the fractions.
 
  • #3
So it's
(loga/log4a) + (log4a/log4a) = (loga/log4a) x (log4a/log4a)

= loga/log4a?
 
  • #4
lionely said:
So it's
(loga/log4a) + (log4a/log4a) = (loga/log4a) x (log4a/log4a)

= loga/log4a?

No. How'd you get from the '+' to the 'x'?

Are you getting confused by the log rule that says: [itex]\log m + \log n = \log mn[/itex]? Because that's something quite different.

Just do the addition like normal fraction expressions. The numerator of the combined expression should then be simplified using that log rule I mentioned.
 
  • #5
so it's log4a^2/log4a?
 
  • #6
lionely said:
so it's log4a^2/log4a?

Yes. Now observe that [itex]4a^2 = (2a)^2[/itex].

Use [itex]\log m^n = n\log m[/itex] here.

It's already quite close to the form you require. You just need one more trivial trick to introduce [itex]\log 3a[/itex] into the expression. Remember that [itex]\frac{x}{y} = (\frac{x}{z})(\frac{z}{y})[/itex], where z can be anything because it just cancels out. A little bit more manipulation and you'll get the form you require.
 
  • #7
How about (2log2a/log3a) x (log3a/log4a)?
 
  • #8
Oh yes it is that because yz = (log3a/log4a) x (log2a/log3a)and xyz + 1 =( 2log2a/log3a) x (log3a/log4)

and that is equal to 2yz!
 
  • #9
lionely said:
Oh yes it is that because yz = (log3a/log4a) x (log2a/log3a)


and xyz + 1 =( 2log2a/log3a) x (log3a/log4)

and that is equal to 2yz!

Yup, you got it. :smile:
 
  • #10
Thank you!
 
  • #11
=/ My teacher said what I did was wrong... he said it must be done the way he did it... he changed base of x to base of y then he got xy I think. Then changed the base of xy to base of z.

I don't remember exactly what he did i didn't get a chance to transcribe it... but it was something along the lines of what I said above. But I personally believe the question can be worked more than one way..
 
  • #12
lionely said:
=/ My teacher said what I did was wrong...
You have my sympathies.
 
  • #13
What does that mean? I have a bad teacher? =/
 
  • #14
lionely said:
What does that mean? I have a bad teacher? =/

Everything you did is mathematically valid. The only small caveat is that 'a' should be taken to be positive, but this is generally a given in this sort of question. I don't know what your teacher is on about.
 

1. What is meant by "changing the bases of logs"?

Changing the bases of logs refers to the process of rewriting a logarithmic expression in a different base. This is often done in order to simplify the expression or to solve for a variable.

2. How do you change the base of a logarithm?

To change the base of a logarithm, you can use the logarithm change of base formula: logb(x) = loga(x) / loga(b). This means that to change a logarithm from base a to base b, you can divide the logarithm of x in base a by the logarithm of a in base b.

3. Why would you want to change the base of a logarithm?

Changing the base of a logarithm can be useful in simplifying complex expressions or solving equations involving logarithms. It can also make it easier to compare logarithmic values with different bases.

4. Can any base be used when changing the base of a logarithm?

Yes, any base can be used when changing the base of a logarithm. However, the most commonly used bases are base 10 (log10) and base e (ln or loge).

5. Are there any rules or restrictions when changing the base of a logarithm?

When changing the base of a logarithm, there are a few rules to keep in mind. First, the base of the logarithm cannot be negative, as logarithms of negative numbers are undefined. Second, the value inside the logarithm (the argument) must be positive. Finally, when solving equations involving logarithms, it is important to check for extraneous solutions that may arise when changing the base.

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