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Simplify log(40.5) base 3 - log(10.5) base 3

Mathematics

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6 steps

  1. Apply the logarithm property log⁡b(x)−log⁡b(y)=log⁡b(xy)\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right).

    The given expression is log⁡3(40.5)−log⁡3(10.5)\log_3(40.5) - \log_3(10.5).

    log⁡3(40.5)−log⁡3(10.5)=log⁡3(40.510.5)\log_3(40.5) - \log_3(10.5) = \log_3\left(\frac{40.5}{10.5}\right)

  2. Simplify the fraction inside the logarithm.

    To simplify 40.510.5\frac{40.5}{10.5}, multiply the numerator and denominator by 10 to remove the decimals.

    40.510.5=40.5×1010.5×10=405105\frac{40.5}{10.5} = \frac{40.5 \times 10}{10.5 \times 10} = \frac{405}{105}

    Divide both the numerator and the denominator by their greatest common divisor. Both are divisible by 5.

    405÷5105÷5=8121\frac{405 \div 5}{105 \div 5} = \frac{81}{21}

    Both 81 and 21 are divisible by 3.

    81÷321÷3=277\frac{81 \div 3}{21 \div 3} = \frac{27}{7}

  3. Substitute the simplified fraction back into the logarithm.

    log⁡3(277)\log_3\left(\frac{27}{7}\right)

  4. Apply the logarithm property log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y).

    log⁡3(277)=log⁡3(27)−log⁡3(7)\log_3\left(\frac{27}{7}\right) = \log_3(27) - \log_3(7)

  5. Evaluate log⁡3(27)\log_3(27).

    Since 27=3327 = 3^3, we have log⁡3(27)=3\log_3(27) = 3.

  6. Substitute the value of log⁡3(27)\log_3(27) into the expression.

    3−log⁡3(7)3 - \log_3(7)

    Assuming the notation log⁡37\log_3^7 in the options means log⁡3(7)\log_3(7), the simplified expression matches option C.

    The final answer is \boxed{\text{3 - \log_3^7}}.

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