Evaluating logarithms using change-of-base formula

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  1. How to apply `` change-of-base rule""

    Express log53 \log_53 in three different ways.
    1. Using a calculator, evaluate the following logarithms
      by applying `` change-of-base rule":":
      1. log53\log_53
      2. log7416\log_7\sqrt{416}
      3. log2725\log_2\frac{7}{25}
      4. 6log4999 \log_4999
    2. Using a calculator, solve for x x to the nearest hundredth.
      1. log6x=log78\log_6x = log_7 8
      2. log235=logx0.104\log_{23}5 = log_x\sqrt{0.104}
    Topic Notes
    • change-of-base rule:
    logba=logxalogxb=logalogb \log_ba = \frac{\log_xa}{\log_xb} = \frac{\log a}{\log b}

    • common logarithms:
    log with base 10" ``10"
    example: log3=log103 \log3 = \log_{10}3
    example: logx=log10x \log x = \log_{10}x