Proofs of Logarithm Properties:
Here we learn the general laws of logarithms. First, we list the logarithm formulas:
Proofs of Logarithm Formulas:
Proof of Product Rule of Logarithms
`log_a(MN)=log_a M + log_a N`
Proof:
Suppose that `x=log_a M`
Thus, by the definition of logarithm
`a^x=M`
Again, suppose that `y=log_a N` `Rightarrow a^y=N`
Note that `MN=a^x cdot a^y=a^{x+y}`
Taking logarithms with respect to the base `a` on both sides,
`log_a (MN)=log_a a^{x+y}`
`=x+y` `[because log_a a^k=k]`
`=log_a M + log_a N`
(plug-in the values of x and y)
Hence, `log_a(MN)=log_a M + log_a N` (Proved)
In the same way as above, we can prove
the quotient rule of logarithms.
Proof of Quotient Rule of Logarithms
`log_a(MN)=log_a M + log_a N`
Proof:
Suppose that `x=log_a M`
Thus, by the definition of logarithm
`a^x=M`
Again, suppose that `y=log_a N` `Rightarrow a^y=N`
Note that `M/N=frac{a^x}{a^y}=a^{x-y}`
Taking logarithms with respect to the base `a` on both sides,
`log_a (M/N)=log_a a^{x-y}`
`=x-y` `[because log_a a^k=k]`
`=log_a M – log_a N`
(plug-in the values of x and y)
Hence, `log_a(M/N)=log_a M – log_a N` (Proved)
Proof of Power Rule of Logarithms
`log_a M^n=n log_a M`
Proof:
Suppose that `x=log_a M^n`
Thus, `a^x=M^n`—-(i)
Again, let `y=log_a M`
This implies that `a^y=M` —-(ii)
Combining (i) and (ii) we get that
`a^x=M^n=(a^y)^n` [`because a^y=M`]
`Rightarrow a^x=a^{ny}`
Equating the powers of `a`, we obtain
`x=ny`
Putting the values of x and y, we have
`log_a M^n=n log_a M` (Proved)
Proof of Base Change Rule of Logarithms
`log_a M=log_b M times log_a b`
Proof:
Let `x=log_a M`
Thus, `a^x=M`—-(i)
Now, assume that `y=log_b M` `Rightarrow b^y=M`—(ii)
Again, assume `z=log_a b` `Rightarrow a^z=b` —-(iii)
Combining (i) and (ii), we have
`a^x=b^y=(a^z)^b` [`because a^z=b` by (iii)]
`Rightarrow a^x=a^bz`
Equating the powers of a, we obtain
`x=bz`
Putting the values of x, b and z, we get that
`log_a M=log_b M times log_a b` (Proved)
Corollary: `log_b a=frac{1}{log_a b}`
Proof: Take `M=a` in the above base change formula.
We have
`log_a a=log_b a times log_a b`
`Rightarrow 1=log_b a times log_a b` [`log_a a=1`]
Hence, it follows that
`log_b a=frac{1}{log_a b}` (Proved)
Logarithm Formulas List:
1. `log_a 1=0`
2. `log_a a=1`
3. `a^{log_a M}=M`
4. `log_a (MN)=log_a M + log_a N`
5. `log_a (M/N)=log_a M – log_a N`
6. `log_a M^n=nlog_a M`
7. `log_a M=log_b M times log_a b`
8. `log_b a=1/log_a b`
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