Properties and Operations of Indices

Properties of Indices

• Product of Powers: ( a^m \times a^n = a^{m+n} )
• Quotient of Powers: ( \frac{a^m}{a^n} = a^{m-n} ) (where ( a \neq 0 ))
• Power of a Power: ( (a^m)^n = a^{mn} )
• Power of a Product: ( (ab)^n = a^n \times b^n )
• Power of a Quotient: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} ) (where ( b \neq 0 ))
• Zero Exponent: ( a^0 = 1 ) (where ( a \neq 0 ))
• Negative Exponent: ( a^{-n} = \frac{1}{a^n} ) (where ( a \neq 0 ))

Operations with Indices

1. Multiplication:

   • Combine the same base: ( a^m \times a^n = a^{m+n} )

2. Division:

   • Subtract exponents: ( \frac{a^m}{a^n} = a^{m-n} )

3. Raising to a Power:

   • Multiply the exponents: ( (a^m)^n = a^{mn} )

4. Adding and Subtracting Indices:

   • Only combine like bases: ( a^m + a^m = 2a^m ), not ( a^{m+m} )

5. Mixed Operations:

   • Follow the order of operations (PEMDAS/BODMAS) when combining indices.

Simplifying Expressions with Indices

• Identify Common Bases: Look for common bases to apply properties of indices.
• Convert Negative Exponents: Change negative exponents to positive by taking reciprocals.
• Combine Like Terms: For terms with the same base and exponent, combine coefficients.
• Factorization: Factor expressions where possible using indices properties.
• Use of Parentheses: Be cautious with operations involving parentheses and powers to avoid errors.

Example Simplifications

• ( a^3 \times a^2 = a^{5} )
• ( \frac{b^4}{b^2} = b^{2} )
• ( (x^2)^3 = x^{6} )
• ( \left(2^3\right)^2 = 2^{6} = 64 )
• ( 4x^{-2} + 2x^{-2} = 6x^{-2} ) (can also be written as ( \frac{6}{x^2} ))

Properties of Logarithms

• Product Property states that the logarithm of a product equals the sum of the logarithms: ( \log_b(MN) = \log_b(M) + \log_b(N) ).
• Quotient Property indicates that the logarithm of a quotient equals the difference of the logarithms: ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) ).
• Power Property explains that the logarithm of a number raised to a power is the power times the logarithm of the number: ( \log_b(M^k) = k \cdot \log_b(M) ).
• The logarithm of 1 is always zero, regardless of the base, expressed as ( \log_b(1) = 0 ).
• The logarithm of the base is always one, which is written as ( \log_b(b) = 1 ).
• Change of Base Formula allows conversion of logarithms to a different base: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} ), applicable for any positive base ( k ).

Logarithmic Equations

• To solve for ( x ) in ( \log_b(x) = y ), it can be rewritten in exponential form as ( x = b^y ).
• Logarithmic equations involving addition can be combined: ( \log_b(x) + \log_b(y) = \log_b(xy) ).
• Logarithmic equations involving subtraction can be expressed as a quotient: ( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) ).
• An example is provided with ( \log_2(x - 3) = 5 ), leading to the solution ( x = 35 ) after transformation into ( x - 3 = 2^5 ).

Changing Bases

• Changing logarithmic bases is essential, particularly for calculations using calculators that primarily compute base 10 or ( e ).
• The conversion follows the same formula as the Change of Base Formula: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} ), where ( k ) is often 10 or ( e ).
• As an example, converting ( \log_2(8) ) to base 10 demonstrates the process: ( \log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)} ).

Natural Logarithm and e

• Natural logarithm refers to logarithms with base ( e ), represented as ( \ln(x) ).
• The value of ( e ) is approximately 2.71828, which is a significant mathematical constant.
• Properties of natural logarithms include:
  • ( \ln(e) = 1 )
  • ( \ln(1) = 0 )
  • ( \ln(xy) = \ln(x) + \ln(y) )
  • ( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) ).
• In exponential terms, if ( \ln(x) = y ), then it follows that ( e^y = x ).