logarithm bases: Let log(subscript)aX=c, log(subscript)bX=d

[mhw0618]

let log(subscript)aX=c and log(subscript)bX=d find the general statement that expresses log(subscript)abX in terms of c and d

test the validity of your general statement using other values of a, b, and x

discuss the scope and/or limitations of a, b, and x

explain how you arrived at your general statement

 

[stapel]

let log(subscript)aX=c and log(subscript)bX=d find the general statement that expresses log(subscript)abX in terms of c and d

I will guess that you mean the following:

. . . . .log[sub:1lq0z5gf]a[/sub:1lq0z5gf](x) = c

. . . . .log[sub:1lq0z5gf]b[/sub:1lq0z5gf](x) = d

..and that you need to restate log[sub:1lq0z5gf]ab[/sub:1lq0z5gf](x) in terms of c and d. (Otherwise, we need information on how X relates to x and the other variables.)

What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

Eliz.

 

[CirrosiN]

Can anybody answer this? I have a similar problem and would like to see the aforementioned one worked out.

 

[mmm4444bot]

Can anybody answer this? I have a similar problem and would like to see the aforementioned one worked out.

Hello CirrosiN:

Yes, I believe that there may actually be more than one million people on this planet who can answer this.

The overall principle at this web site is:

1) you try to do the exercise yourself

2) you show us any work that you are able to accomplish

3) you try to say something about why you're stuck

4) we respond by guiding you

There is a post at the top of each board's index titled something like, "Read This Before You Post".

Please, read it.

Cheers,

~ Mark

 

[CirrosiN]

Well, in the above problem, I'm stuck with the fact that there are no numerical values given, just variables. I get stuck in the variables and can find no similarities that can connect the two other than the presence of x. I know that a^c = x and that b^d = x but am not sure how to create a general statement based off of that alone.

 

[stapel]

Well, in the above problem, I'm stuck with the fact that there are no numerical values given....

Um... no, you aren't given values; you're told to pick various sets of values, and then see what patterns you can discern. Are you saying that you need help in thinking up numbers...? :shock:

Eliz.

 

[Deleted member 4993]

Well, in the above problem, I'm stuck with the fact that there are no numerical values given, just variables. I get stuck in the variables and can find no similarities that can connect the two other than the presence of x. I know that a^c = x and that b^d = x but am not sure how to create a general statement based off of that alone.

b^d = x

b^c = x^(c/d)

a^c = x

a^c * b^c = x * x^(c/d)

Now continue.....

 

[PAULK]

Can anybody answer this? I have a similar problem and would like to see the aforementioned one worked out.

You asked for it -- you'd better like it.
I'll use this (admittedly sloppy notation) for the logarithm-exponential duality. (I got that term from a Stardrek episode.)

loga X = c means a^c = x
logb x = d means b^d = x
and
logab X = Y, for some Y, which we don't have, and means (ab)^Y = X,

But (ab)^Y = a^y b^y, and X = a^c or b^d, so

a^Y b^Y = a^c or = b^d

Since a^c = X = b^d, we can write

a = b^(d/c) or b = a^(c/d)

a^Y b^Y = a^c << use this one.
a^Y (a^(c/d))^Y = a^c
a^Y a^(cY/d) = a^c

Now just solve for y:
y + cy/d = c

y(1 + c/d) = c
c
Y = --------
1 + c/d

cd
Y = -------
d + c
Same thing either way -- i.e. the log(base ab) is the product of the logs over the sum of the logs. [Amazing.]

Ex:

log2(1024) = 10
log4(1024) = 5
log8(1024) = 10/3


10(5) 50
---- = ---- = 10/3
10+5 15