Laws of Logarithms

Laws of Logarithms

Logarithms are the inverse operations of exponents. If:

${\log }_{b}x=y\text{then}{b}^y=x$

where 

b is the base, 

x is the number, and 

y is the logarithm.

The fundamental laws of logarithms are as follows:

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1. Product Law

${\log }_b(MN)={\log }_{b}M+{\log }_{b}N$

Example:

${\log }_2(8\times 4)={\log }_{2}8+{\log }_{2}4$

Since 

${\log }_{2}8=3$

and 

${\log }_{2}4=2$

 we get:

${\log }_{2}32=3+2=5$

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2. Quotient Law

${\log }_b\left(\frac{M}{N}\right)={\log }_{b}M-{\log }_{b}N$

Example:

${\log }_3\left(\frac{81}{9}\right)={\log }_{3}81-{\log }_{3}9$

Since 

${\log }_{3}81=4$${\log }_{3}9=2$

we get:

${\log }_{3}9=4-2=2$

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3. Power Law

${\log }_b(M^n)=n{\log }_{b}M$

Example:

${\log }_5({25}^3)=3{\log }_{5}25$

Since 

${\log }_{5}25=2$

 we get:

${\log }_5({25}^3)=3\times 2=6$

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4. Change of Base Law

${\log }_{b}M=\frac{{\log }_{k}M}{{\log }_{k}b}$

This is useful for converting logarithms into a different base, commonly base 10 (logarithm) or base 

(natural logarithm, denoted as ln).

Example: Convert 

${\log }_{2}10$

${\log }_{2}10=\frac{{\log }_{10}10}{{\log }_{10}2}$

Since 

${\log }_{10}10=1$

and 

${\log }_{10}2\approx 0.301$

, we get:

${\log }_{2}10=\frac{1}{0.301}\approx 3.32$

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5. Logarithm of 1

${\log }_{b}1=0$

Because any number raised to the power of zero equals 1:

$b^0=1$

Example:

${\log }_{7}1=0$

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6. Logarithm of the Base

${\log }_{b}b=1$

Because 

$b^1=b$

Example:

${\log }_{4}4=1$

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7. Natural Logarithms (Base)

$\ln x={\log }_{e}x$

Natural logarithms use the base (Euler’s number, approximately 2.718). The same logarithmic laws apply:

$\ln (MN)=\ln M+\ln N$

$\ln \left(\frac{M}{N}\right)=\ln M-\ln N$$\ln (M^n)=n\ln M$

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Examples Using Multiple Laws

1. Simplify 

   ${\log }_{3}81+{\log }_{3}9$

   • Using the product law:

     ${\log }_3(81\times 9)={\log }_{3}729$

   • Since 

     $3^6=729$

      we get:

     ${\log }_{3}729=6$

2. Simplify 

   ${\log }_{5}125-{\log }_{5}25$

   • Using the quotient law:

     ${\log }_5\left(\frac{125}{25}\right)={\log }_{5}5$

   • Since 

     ${\log }_{5}5=1$

     , the answer is:

     $1$

3. Simplify 

   $2{\log }_{4}8$

   • Using the power law:

     ${\log }_4{8}^2={\log }_{4}64$

   • Since 

     $4^3=64$

      we get:

     ${\log }_{4}64=3$