Can someone prove log (base a)(MN) = (log (base a) M) + (log (base a) N)?

Question

I'm just trying to get a better (more intuitive) comprehension of logarithms. I read a proof, but did not follow.

Answer 1

First, you should try some examples with specific numbers. Like
log_2 8 = log_2 2 + log_2 4.

Then, think about what logarithm means and also, note that when you are multiplying say a^n to a^m, you get a^(n+m).

Answer 2

The word "log" is nothing more than another word for exponent or power. One of the laws of exponents says that when two or more numbers of the same base are multiplied together, the exponent of their product is equal to the sum of the exponents of each factor. Mathematically speaking then:

(a^x)(a^y) = a^(x+y).

When we say the log of one side of an equation is equal to the log of the other side, what we are really saying is that the exponent to the base a on the left hand side is equal to the exponent to the base a on the right hand side.

If we let M = a^x and N = a^y then MN = (a^x )(a^y) = a^(x+y) from our discussion concerning the law of exponents above.

So, if we take the log, or exponent, to the base a of each side, we get:

exp (base a) MN = exp (base a) a^(x+y) = x+y.

Therefore, by transitivity: exp or log (base a) MN = x+y.

You can prove this simple fact to yourself simply by multiplying two numbers of the same base raised to different or the same power (think exponent or log) together, then analyzing their product. Since we use powers of 10 frequently here in this country, let's use them.

(1,000)(100,000) = (10^3 )(10^5).

Look at the left hand side for now. We know from grade school that to multiply these numbers together we simply write down a 1, then write down the sum of their 0's after that to get our product. We note there are eight 0's here, so we write down (1,000)(100,000) = 100,000,000. We also note that 100 million is 10^8, so its log or exponent (base 10) is 8.

We now are ready to equate the product we have calculated on the left hand side with the right hand side of the equation.

100,000,000 = 10^8 = (10^3)(10^5).

Let MN = 100,000,000. Now, we take logs on both sides:

log (MN) = log (100,000,000) = log (10^8) = 8 = log [(10^3)(10^5)] = log [10^(3+5)] = log [10^8] = 8.

So, the log of the product of two numbers to the same base is equal to the sum of their individual logs on the same base number.

Answer 3

When you want to prove Logarithm rules, try to convert logs into some kind of variables.

1)
Log(base a)M = K
Log(base a)N = J

2)
a^K = M
a^J = N

3)
a^K + a^J = M * N
a^(K + J) = MN

4)
Log(base a) MN = K + J

Remember,
K = Log(base a)M
J = Log(base a)N

Thus.

5)
Log(base a) MN = Log(base a) M + Log(base a) N


hope this helped.

Answer 4

I kind of agree with gold, but I suggest getting rid of the base at first, and going with log or ln.

ln (e^16) = ln e^4 + ln e^12

16 ln e = 4 ln e + 12 ln e

ln e = 1

16 = 4 + 12.

See?