Lesson Properties of the logarithm

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Properties of Logarithm



The definition of the logarithm is given in lesson WHAT IS the logarithm in this site.
Now we consider properties of logarithms - formulas for the logarithm of a product, logarithm of a quotient, logarithm of a power and logarithm of a root.

Product Rule


The logarithm of a product of two positive real numbers is equal to the sum of logarithms of factors:
log%28b%2C%28MN%29%29+=+log%28b%2CM%29+%2B+log%28b%2CN%29


Examples
1) According to the Product Rule, log%282%2C%284%2A8%29%29+=+log%282%2C4%29+%2B+log%282%2C8%29+=+2+%2B+3+=+5.
Check it by making the direct calculation: log%282%2C%284%2A8%29%29+=+log%282%2C32%29+=+5. You get exactly the same number as the Product Rule produces!

2) According to the Product Rule, log%2810%2C%280.1%2A1000%29%29+=+log%2810%2C0.1%29+%2B+log%2810%2C1000%29+=+-1+%2B+3+=+2.
Verify it by making the direct calculation: log%2810%2C%280.1%2A1000%29%29+=+log%2810%2C100%29+=+2. You get the same result as the Product Rule produces.

Proof of the Product Rule
Let log%28b%2CM%29=x, log%28b%2CN%29=y.
Then b%5Ex=M, b%5Ey=N due to the logarithm definition (see lesson WHAT IS the logarithm).
Multiplying these two equations, you obtain
MN=b%5Ex%2Ab%5Ey+=+b%5E%28x%2By%29.
This means log%28b%2C%28MN%29%29=+x%2By due to the logarithm definition.
Substitute x=log%28b%2CM%29 and y=log%28b%2CN%29 to the last formula, and you obtain the required formula
log%28b%2C%28MN%29%29+=+log%28b%2CM%29+%2B+log%28b%2CN%29.

Quotient Rule


The logarithm of a quotient of two positive real numbers is equal to the logarithm of the dividend minus the logarithm of the divisor:
log%28b%2C%28M%2FN%29%29+=+log%28b%2CM%29+-+log%28b%2CN%29


Examples
3) According to the Quotient Rule, log%282%2C%288%2F2%29%29+=+log%282%2C8%29+-+log%282%2C2%29+=+3+-+1+=+2.
Check it: log%282%2C%288%2F2%29%29+=+log%282%2C4%29+=+2. You get exactly the same number as the Quotient Rule produces.

4) According to the Quotient Rule, log%2810%2C%281000%2F10%29%29+=+log%2810%2C1000%29+-+log%2810%2C10%29+=+3+-+1+=+2.
Verify it: log%2810%2C%281000%2F10%29%29+=+log%2810%2C100%29+=+2. You get the same result as the Quotient Rule produces.

Proof of the Quotient Rule
Let log%28b%2CM%29=x, log%28b%2CN%29=y.
Then b%5Ex=M, b%5Ey=N due to the logarithm definition (see lesson WHAT IS the logarithm).
Dividing these two equations, you obtain
M%2FN=b%5Ex%2Fb%5Ey+=+b%5E%28x-y%29.
This means log%28b%2C%28M%2FN%29%29=+x-y due to the logarithm definition.
Substitute x=log%28b%2CM%29 and y=log%28b%2CN%29 to the last formula, and you obtain the required formula
log%28b%2C%28M%2FN%29%29+=+log%28b%2CM%29+-+log%28b%2CN%29.

Power Rule


The logarithm of a power of positive number is equal to the exponent times the logarithm of the number:
log%28b%2C%28M%5Ep%29%29+=+p%2Alog%28b%2CM%29


Examples
5) According to the Power Rule for logarithms, log%282%2C%288%5E2%29%29+=+2%2Alog%282%2C8%29=+2%2A3+=+6.
Check it: log%282%2C%288%5E2%29%29+=+log%282%2C64%29+=+6. You get the same number as the Power Rule produces.

6) According to the Power Rule, log%2810%2C%281000%5E3%29%29+=+3%2Alog%2810%2C1000%29+=+3%2A3+=+9.
Check it: log%2810%2C%281000%5E3%29%29+=+log%2810%2C1000000000%29+=+9. You get the same result as the Power Rule produces.

Proof of the Power Rule
Let log%28b%2CM%29=x.
Then b%5Ex=M due to the logarithm definition (see lesson WHAT IS the logarithm).
Raising both sides to the p-th power, you obtain
M%5Ep=%28b%5Ex%29%5Ep+=+b%5E%28p%2Ax%29.
This means log%28b%2C%28M%5Ep%29%29=+p%2Ax due to the logarithm definition.
Substitute x=log%28b%2CM%29 to the last formula, and you obtain the required formula
log%28b%2C%28M%5Ep%29%29+=+p%2Alog%28b%2CM%29.

Logarithm of a Root


The logarithm of a root of positive number is equal to the logarithm of the number divided by the index of the root:
log%28b%2Croot%28m%2Cp%29%29+=+%281%2Fm%29%2Alog%28b%2Cp%29


Examples
7) According to the Root Formula for logarithms, log%282%2Csqrt%283%29%29+=+%281%2F2%29%2Alog%282%2C3%29.
Compare it with what the Power Rule produces: log%282%2Csqrt%283%29%29+=+log%282%2C%283%5E%281%2F2%29%29%29+=+%281%2F2%29%2Alog%282%2C3%29. The Root Formula produces exactly the same number as the Power Rule.

8) According to the Root Formula, log%2810%2Csqrt%282%29%29+=+%281%2F2%29%2Alog%2810%2C2%29.
Compare it with what the Power Rule produces: log%2810%2Csqrt%282%29%29+=+log%2810%2C%282%5E%281%2F2%29%29%29+=+%281%2F2%29%2Alog%2810%2C2%29. The Root Formula produces exactly the same result as the Power Rule.

The Root Formula is a special case of the Power Rule and therefore doesn't require the separate proof.

Couple examples below illustrate how to use logarithm properties together.

9) Calculate log%28b%2C%28%28x%5E3%29%2F%28yz%29%29%29.
log%28b%2C%28%28x%5E3%29%2F%28yz%29%29%29+=+log%28b%2C%28x%5E3%29%29-log%28b%2C%28yz%29%29+=+3%2Alog%28b%2Cx%29-%28log%28b%2Cy%29%2Blog%28b%2Cz%29%29%29.

10) Calculate log%28b%2Croot%284%2C%28x%2Ay%29%2F%28z%5E3%29%29%29.
log%28b%2Croot%284%2C%28x%2Ay%29%2F%28z%5E3%29%29%29+=+%281%2F4%29%2Alog%28b%2C%28%28x%2Ay%29%2F%28z%5E3%29%29%29+=+%281%2F4%29%2A%28log%28b%2Cx%29%2Blog%28b%2Cy%29-3%2Alog%28b%2Cz%29%29.

Summary


log%28b%2C%28MN%29%29+=+log%28b%2CM%29+%2B+log%28b%2CN%29
log%28b%2C%28M%2FN%29%29+=+log%28b%2CM%29+-+log%28b%2CN%29
log%28b%2C%28M%5Ep%29%29+=+p%2Alog%28b%2CM%29
log%28b%2Croot%28m%2Cp%29%29+=+%281%2Fm%29%2Alog%28b%2Cp%29

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