Q:
# How Do You Add Logarithms?

**Logarthims of the same base can be added together by multiplying their arguments and then performing the logarithm on the product.** For example, assuming log means log base 10 as it does on a calculator:
log(x) + log(y) = log(x * y)

Applying the definition of logarithms, we can see why this property is true. To solve log(x) = z, we do the following algebra:

10^z = x

Similarly, to solve log(y) = w, we do:

10^w = y

So, multiplying the two expressions together, we get:

(10^z)(10^w) = xy

or, 10^(z+w) = xy

Now, going in reverse, we see that this is a logarithmic expression equivalent to log(xy)

Applying the property to a problem with real numbers:

log(5) + log(3) = log(3 * 5) = log(15)

The property mentioned above is applicable to all logarithms. For example, for the natural logarithm, ln (also defined as log base e), the following is true:

ln(4) + ln(7) = ln(4*7) = ln(28).

Note that logarithms of different bases cannot be combined. For example, the following statements are both false:

ln(3) + log(9) = log(3*9)

ln(3) + log(9) = ln(3*9)

Similarly, for subtraction, the following property holds:

log(x) - log(y) = log(x/y) = log(2)

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