Exponents and Logarithms

Properties Of Exponents

• Product of Powers: ( am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n)
• Quotient of Powers: ( aman=am−n)(for(a≠0)\frac{a^m}{a^n} = a^{m-n} ) (for ( a \neq 0 )anam​=am−n)(for(a=0))
• Power of a Power: ( (am)n=amn(a^m)^n = a^{mn}(am)n=amn )
• Power of a Product: ( (ab)m=am⋅bm(ab)^m = a^m \cdot b^m(ab)m=am⋅bm )
• Power of a Quotient: ( (ab)m=ambm)(for(b≠0)\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} ) (for ( b \neq 0 )(ba​)m=bmam​)(for(b=0))
• Zero Exponent: ( a0=1a^0 = 1a0=1 ) (for ( a≠0a \neq 0a=0 ))
• Negative Exponent: ( a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​ ) (for ( a≠0a \neq 0a=0 ))

Logarithmic Functions

• Definition: If ( by=xb^y = xby=x ), then ( log⁡b(x)=y\log_b(x) = ylogb​(x)=y )
• Common Bases:
  • Base 10: ( log⁡10(x)\log_{10}(x)log10​(x) ) or ( log⁡(x)\log(x)log(x) )
  • Base ( e ): ( ln⁡(x)\ln(x)ln(x) ) (natural logarithm)
• Properties:
  • Product Rule: ( log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)logb​(xy)=logb​(x)+logb​(y) )
  • Quotient Rule: ( log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)logb​(yx​)=logb​(x)−logb​(y) )
  • Power Rule: ( log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n \log_b(x)logb​(xn)=nlogb​(x) )
  • Change of Base: ( log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}logb​(x)=logk​(b)logk​(x)​ ) for any base ( k )

Change Of Base Formula

• Allows calculation of logarithms with different bases:
  • ( log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}logb​(x)=logk​(b)logk​(x)​ )
  • Commonly used with ( k = 10 ) or ( k = e )
• Simplifies complex logarithmic calculations using more familiar logarithmic bases.

Solving Exponential Equations

• To solve equations of the form ( ax=ba^x = bax=b ):

  • Take the logarithm of both sides: ( log⁡(ax)=log⁡(b)\log(a^x) = \log(b)log(ax)=log(b) )
  • Use properties of logarithms: ( xlog⁡(a)=log⁡(b)x \log(a) = \log(b)xlog(a)=log(b) )
  • Solve for ( x ): ( x=log⁡(b)log⁡(a)x = \frac{\log(b)}{\log(a)}x=log(a)log(b)​ )

• For equations like ( ax=aya^x = a^yax=ay ):

  • If the bases are the same, then ( x = y ).

Applications Of Logarithms

• Exponential Growth/Decay: Modeling populations, radioactive decay, and interest calculations using the form ( y=abty = ab^ty=abt ).
• pH Scale: A logarithmic scale for acidity: ( pH=−log⁡[H+]\text{pH} = -\log[H^+]pH=−log[H+] ).
• Richter Scale: Measures earthquake magnitudes logarithmically.
• Sound Intensity: Decibels (dB) are often calculated using logarithmic scales to quantify sound intensity: ( dB=10log⁡(II0)\text{dB} = 10 \log\left(\frac{I}{I_0}\right)dB=10log(I0​I​) ).
• Data Scaling: Logarithms help in normalizing data, especially in statistics and machine learning applications.