Can you suggest why the property of log x + log y = log (xy) is true?

Question

I cannot seem to verbalize this into words. Why can this property be proved true?

Answer 1

The actual proof of this log property (along with the other log properties) stems from these exponential facts:

x^(a) * x^(b) = x^(a+b)
x^(a) / x^(b) = x^(a - b)
(x^a)^b = x^(ab)

Let's start our proof.

Claim: log [base b](ac) = log[base b](a) + log[base b](c)

Let x = log[base b](a) and y = log[base b](c). Converting both to those to logarithmic form, we get

b^x = a, and b^y = c

Now, let's multiply the equations together; that is, the left hand sides with the right hand sides. As a result, we get

(b^x)(b^y) = ac

Now, using the exponential property of exponents with the same base,

b^(x + y) = ac

Now, let's convert this to logarithmic form.

log[base b](ac) = x + y

Substituting back x and y, we get

log[base b](ac) = log[base b](a) + log[base b](c)

Answer 2

Because logs are exponents, and, when we multiply, we add exponents. If you want to put it into words, use easy numbers with log base 2, and x and y easy powers of 2, for example, 4 and 8, or 4 and 16.
So with log base 2, x = 4, y = 8, (I'm going to suppress the base to make this easier to read, base 2 is understood all across)
log 4 + log 8 = 2 + 3 = 5 = log (4x8) = log 32 = 5
Try it with some more numbers like this and convince yourself that it's true.

Answer 3

Examples are the best teacher. log 100 + log 1000 = 2+3 = 5
log (100*1000) = log (100000) = 5
The log is the exponant - when you multiply two of the same base with different exponants you add them.
10^2 * 10^3 = 10^5 = 100000 = 100*1000

Answer 4

OK, suppose the log is in base b, and let a = logx + logy. Taking each side as a power of the base, we have

b^a = b^(logx + logy)
b^a = b^logx * b^logy
b^a = xy

take the log of both sides:
a = log(xy).

Answer 5

The rule of multiplication is
x^a1 * x^a2 * x^a3 *...... x^an= x^(a1+a2+....an) so
for
10^a*10^b=10^(a+b)
take the Log for the last formula, gives
a+b=(a+b) then for any base
log(a*b)=log(a)+log(b)

Answer 6

let log x=a
10^a=x
let log y be b
10^b=y
xy=10^a*10^b
=10^(a+b)
log xy=log 10^(a+b)
=>a+b
=logx+logy