5.05 Logarithm laws
Ideas
Logarithm laws
In the same way that there are index laws which allow us to simplify exponential expressions, there are logarithm laws that allow us to simplify logarithmic expressions. In fact, each logarithm law is a consequence of an index law.
First, consider the definition that if y=B^x then x=\log_{B}y. It follows that,\log_{B} B^x=xSubstituting x=0 and x=1 gives the following special cases:\log_{B} 1=0 \\ \log_{B} B=1
For the following proofs, we will let p=B^m and q=B^n so that \log_{B}p=m and \log_{B}q=n. Note that for any B \neq 0 there will be some values of m, \, n, \, p, and q which makes these equations true.
| \displaystyle B^m \times B^n | \displaystyle = | \displaystyle B^{m+n} | Multiplication law |
| \displaystyle \log_{B} (B^m \times B^n) | \displaystyle = | \displaystyle \log_{B} B^{m+n} | Take the logarithm of both sides |
| \displaystyle \log_{B} (B^m \times B^n) | \displaystyle = | \displaystyle m+n | Apply \log_{B} B^x=x |
| \displaystyle \log_{B} pq | \displaystyle = | \displaystyle \log_{B}p + \log_{B}q | Substitute p=B^m and q=B^n |
| \displaystyle \dfrac{B^m}{B^n} | \displaystyle = | \displaystyle B^{m-n} | Power of a power law |
| \displaystyle \log_{B} \dfrac{B^m}{B^n} | \displaystyle = | \displaystyle \log_{B} B^{m-n} | Take the logarithm of both sides |
| \displaystyle \log_{B} \dfrac{B^m}{B^n} | \displaystyle = | \displaystyle m-n | Apply \log_{B} B^x=x |
| \displaystyle \log_{B} \dfrac{p}{q} | \displaystyle = | \displaystyle \log_{B}p - \log_{B}q | Substitute p=B^m and q=B^n |
| \displaystyle \left(B^m\right)^n | \displaystyle = | \displaystyle B^{mn} | Powere of a power law |
| \displaystyle \log_{B} \left(B^m\right)^n | \displaystyle = | \displaystyle \log_{B} B^{mn} | Take the logarithm of both sides |
| \displaystyle \log_{B} \left(B^m\right)^n | \displaystyle = | \displaystyle mn | Apply \log_{B} B^x=x |
| \displaystyle \log_{B} p^{n} | \displaystyle = | \displaystyle n \log_{B}p | Substitute p=B^m |
Examples
Example 1
Simplify \dfrac{\log_{4}49}{\log_{4}7}.
Example 2
Simplify \log_{}2x + \log_{}50y to an expression with a single logarithmic term.
The definition of logarithms and the index laws give us the following results:
| \displaystyle \log_{B} B^x | \displaystyle = | \displaystyle x |
| \displaystyle \log_{B} 1 | \displaystyle = | \displaystyle 0 |
| \displaystyle \log_{B} B | \displaystyle = | \displaystyle 1 |
| \displaystyle \log_{B} pq | \displaystyle = | \displaystyle \log_{B}p + \log_{B}q |
| \displaystyle \log_{B} \dfrac{p}{q} | \displaystyle = | \displaystyle \log_{B}p - \log_{B}q |
| \displaystyle \log_{B} p^{n} | \displaystyle = | \displaystyle n \log_{B}p |