Precalculus - Logarithm Laws

10 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Logarithmic form

   log⁡bx=a\log_{b}{x}=alogb​x=a

2. Exponential form

   x=bax=b^ax=ba

3. Exponential form to Logarithmic form

   (x=ba)⇒(log⁡bx=a)(x=b^a)\Rightarrow (\log_{b}{x}=a)(x=ba)⇒(logb​x=a)

4. Natural Logarithm

   ln⁡x=log⁡ex\ln x=\log_{e}{x}lnx=loge​x

5. 0\wedge M,N\in \mathbb{R}, \text{ and } b>1\wedge b\in\mathbb{R}\text{, then:}"> if M,N>0∧M,N∈R, and b>1∧b∈R, then:\text{if } M,N>0\wedge M,N\in \mathbb{R}, \text{ and } b>1\wedge b\in\mathbb{R}\text{, then:}if M,N>0∧M,N∈R, and b>1∧b∈R, then:

   Used in the following flashcards

6. Multiplication inside logarithm

   log⁡b(M⋅N)=log⁡bM+log⁡bN\log_{b}(M\cdot N)=\log_{b}M + \log_{b}Nlogb​(M⋅N)=logb​M+logb​N

7. Division inside logarithm

   log⁡b(MN)=log⁡bM−log⁡bN\log_{b}(\frac{M}{N})=\log_{b}M - \log_{b}Nlogb​(NM​)=logb​M−logb​N

8. Logarithm Equality

   log⁡bM=log⁡bN if and only if M=N \log_{b}M=\log_{b}N \text{ if and only if } M=Nlogb​M=logb​N if and only if M=N

9. Logarithm of exponents

   log⁡bMk=klog⁡bM\log_{b}M^k=k\log_b{M}logb​Mk=klogb​M

10. Base to a logarithmic exponent

    blog⁡bM=Mb^{\log_b{M}}=Mblogb​M=M