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log⁡863−log⁡89=\log_{8}63-\log_{8}9=

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Q. log⁡863−log⁡89=\log_{8}63-\log_{8}9=
  1. Understand Logarithm Properties: Understand the properties of logarithms. We can use the quotient rule for logarithms, which states that log⁡b(M)−log⁡b(N)=log⁡b(MN)\log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right), where bb is the base of the logarithms.
  2. Apply Quotient Rule: Apply the quotient rule to the given expression. log⁡863−log⁡89\log_{8}63 - \log_{8}9 becomes log⁡8(639)\log_{8}(\frac{63}{9}).
  3. Simplify Fraction: Simplify the fraction inside the logarithm. \newline639\frac{63}{9} simplifies to 77.\newlineSo, log⁡863−log⁡89\log_{8}63 - \log_{8}9 becomes log⁡87\log_{8}7.
  4. Evaluate Logarithm: Evaluate log⁡87\log_{8}7. Since 77 cannot be expressed as a power of 88, the logarithm does not simplify to a nice whole number or fraction. The value of log⁡87\log_{8}7 is not an integer or a simple fraction, and it is typically left as is or approximated using a calculator.

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