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.

This quiz covers essential properties of exponents and logarithmic functions, including definitions and key rules. Test your understanding of how to manipulate powers and logarithms effectively. Perfect for mastering foundational algebra concepts!