multiplying logs (see question posed below?

Question

I dont get how multiplying logs means you add them, i just do not grasp the idea.

for example (lets take logs to base 10)

log 10 * log 10 = 1

log 10 + log 10 = Log 100 = 2

The answer is not the same, so I just dont understand how

log3 x ylog5 be comes log3 + ylog 5

I know that

log a - log b = log (a/b)
log A + log b = log ab
log a^b = bloga.

Am i missing something glaringly obvious?

Answer 1

log 10 * log 10 = 1 because log (base 10) 10 = 1
and therefore 1 * 1 = 1

log 10 + log 10 = 1 + 1 = 2
and log 10 + log 10 = log (10*10) = log 100
log 100 = 2 means 10^2 = 100 noting again that log means log (base 10) meaning it is the power to which the base is raised to get its argument

1000 = 10^3 so log 1000 = 3

Now using the rules stated
[ie log a - log b = log (a/b)
log a + log b = log ab
log a^b = bloga]

log 3 + 3 log y = log3 + log (y^5) (since bloga = log a^b)
= log(3 * y^5) (since log a + log b = log ab)
= log (3y^5)

The log of a product = the SUM of the individual logarithms!

log(ab) = loga PLUS logb defintely NOT loga TIMES log b

Answer 2

Logarithm when it is written as "log" ((by itself)) always means base ten. If it is not base ten, it is written as "log( base)" then the number, or natural logarithms base e are written "ln".
In any case the log (10) of 10 is 1, Therefore, log 10 time log 10 equall 1 time 1 or 1.
The log(base x) of x is 1.
Likewise, the log(base x) of x squared is 2.
So, log(2) of 2 times log(2) of 2 is 1,
And, log(2) of 2 plus the log(2) of 2 is 2..
The key is knowing the base of "log 10" is 10.

Answer 3

ok, they are using log base 10! this is important because when we write log now we usually mean log base "e."

the log 10 when the base is 10 is equal to one. Think of 10^x=10 the expotent you would want to satisfy this equation is x=1. This is what is meant by log. The first question then is just log10*log10=1*1=1

the next one you are adding. So, log10+log10 = 1 PLUS 1 = 2.

The difference is in multiplying one by itself and adding one to itself.

Answer 4

log 10 + log 10 = Log 100 = 2
Correct - adding these logs corresponds to multiplying 10*10.

log 10 * log 10 = 1
Multiplying logs does NOT correspond to multiplying 10*10.
It corresponds to exponentiating one number to the power of the second log i.e. it's:
log a^b = bloga.
where a = 10, b = ** log 10 ** (note: not 10 !)

So log 10 * log 10 It corresponds with raising 10 to the exponent log10=1

Answer 5

EDIT: I saw your previous question. If that is indeed "log3 x ylog5" it is definitely a typo. It should be

log(3*5^y) =
log3 + log(5^y) =
log3 + ylog5.