How does one multiply two logarithms together? Is it possible?

Question

I understand that the addition of two logarithms means that the numbers should be multiplied. For example:
log 3 + log 4 = log 12

However, I don't understand what to do when the logarithms need to be multiplied together (instead of the addition becoming a multiplication)?
Can this be done?
For example:
log 25 multiplied by log 3 (base of 10).

I've been messing around with my calculator and can't seem to get it.

Also... what happens when multiplying with different bases? Can anyone tell me how that works?

Thanks in advance!
(Note: I'm not looking for the answer in particular - the explanation is more important.)

Answer 1

Just remember this: alog(b) = log(b^a)



So log(a)log(b) = log[b^log(a)] = log[a^log(b)]



But unless you have an easy expression for them I'd leave it as the product of logs.



This presumes they have the same base!



If you have logs with different bases then you have to convert them.



Denote log_b(a) as the log of a under base b



log_b(a) = log_c(a)/log_c(b) <-- forumla for change of base



So if you have log_b(a) * log_c(k) then the result is



log_b(a) * log_c(k) = [log_c(a)/log_c(b)] * log_c(k)



log_b(a) * log_c(k) = log_c(a)log_c(k)/log_c(b)



And this is really as simplified as you're going to get.

Answer 2

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ln (4x - 3) + ln (x - 5) = ln 15 (combine the natural logs) ln ((4x - 3)(x - 5)) = ln 15 (drop the natural logs) (4x - 3)(x - 5) = 15 (FOIL) 4x^2 - 23x + 15 = 15 (subtract 15 from both sides) 4x^2 - 23x = 0 (factor out x) x(4x - 23) = 0 (zero-product property) x = 0 or 4x - 23 = 0 x = 0 or 4x = 23 x = 0 or x = 23/4 The possible solutions of the equation are 0 and 23/4, however when you substitute 0 for x into the equation, you end up taking the log of a negative, which is not possible, so x cannot equal 0. Therefore x = 23/4 is your solution. ANSWER: x = 23/4 Remember that logs of the same base can be combined by multiplying them together. So for example, log 4 + log 7 = log 28. For your future reference, if they are being subtracted then you can combine them into one log by dividing, like log 6 - log 3 = log 2. You can only drop the logs in the equation if it is log = log. In other words two logs of the same base directly equal each other. If there are other terms, you cannot drop the logs, like if you have log 5 + log x = log 10. Once you combine the left side into one log, which would be log 5x, then you can drop them and you'd end up with 5x = 10 and x = 2.

Answer 3

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RE:

How does one multiply two logarithms together? Is it possible?

I understand that the addition of two logarithms means that the numbers should be multiplied. For example:

log 3 + log 4 = log 12



However, I don't understand what to do when the logarithms need to be multiplied together (instead of the addition becoming a multiplication)?

Can this be...

Answer 4

Multiplying Logs

Answer 5

The sum of logarithms is the logarithm of a product, so ln(4x-3) + ln(x-5) = ln[(4x-3)(x-5)] = ln(15). This can be rewritten as ln(4x² - 23x + 15) = ln(15). To "undo" a logarithm, use its inverse (the exponential): e^ln(4x² - 23x + 15) = e^ln(15), or 4x² - 23x + 15 = 15, which gives 4x² - 23x = 0, which gives x(4x - 23) = 0, which means that x = 0 or x = 23/4. x=0 is not allowed here, since it would mean using a negative argument [such as 4(0)-3, or 0-5] for the logarithm. So, x can only be 23/4.

Answer 6

Of course it is "possible", but there is no special rule for it, like you get when adding logarithms together.



You may be able to rearrange an expression like this in other ways, however, like:

log(6) * log(2)

= log(3*2) * log(2)

= (log(3)+log(2)) * log(2)

= log(3)*log(2) + (log(2))^2

...But as you can see, this does not usually get you anywhere useful, so you would not usually need to do anything like this. Just bear in mind that rearranging may be necessary for "show that..." questions in an exam.

Answer 7

Log A Log B

Answer 8

These videos may help you ALOT



http://www.youtube.com/watch?v=pP3NunYYhzk

http://www.geocities.com/Area51/Shuttle/4899/notes/inl3.html

http://www.vias.org/calculus/08_exp-log_functions_02_03.html



But this website will explain if you dont understand still:

http://www.sosmath.com/algebra/logs/log4/log43/log43.html