How do you add logarithms?


Quick Answer

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)

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Full Answer

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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