Evaluating logarithms using change-of-base formula

Evaluating logarithms using change-of-base formula

Basic Concepts: Converting from logarithmic form to exponential form, Evaluating logarithms without a calculator, Common logarithms
Related Concepts: Logarithmic scale: Richter scale (earthquake), Logarithmic scale: pH scale, Logarithmic scale: dB scale

Lessons

• change-of-base rule:
log⁡ba=log⁡xalog⁡xb=log⁡alog⁡b \log_ba = \frac{\log_xa}{\log_xb} = \frac{\log a}{\log b}

• common logarithms:
log with base ‘‘10" ``10"
example: log⁡3=log⁡103 \log3 = \log_{10}3
example: log⁡x=log⁡10x \log x = \log_{10}x
  • 1.
    How to apply ‘‘ `` change-of-base rule""

    Express log⁡53 \log_53 in three different ways.
  • 2.
    Using a calculator, evaluate the following logarithms
    by applying ‘‘ `` change-of-base rule":":
    a)
    log⁡53\log_53

    b)
    log⁡7416\log_7\sqrt{416}

    c)
    log⁡2725\log_2\frac{7}{25}

    d)
    6log⁡4999 \log_4999

  • 3.
    Using a calculator, solve for x x to the nearest hundredth.
    a)
    log⁡6x=log78\log_6x = log_7 8

    b)
    log⁡235=logx0.104\log_{23}5 = log_x\sqrt{0.104}

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42.
Logarithmic Functions
42.1
What is a logarithm?
42.2
Converting from logarithmic form to exponential form
42.3
Evaluating logarithms without a calculator
42.4
Common logarithms
42.5
Natural log: ln
42.6
Evaluating logarithms using change-of-base formula
42.7
Converting from exponential form to logarithmic form
42.8
Solving exponential equations with logarithms
42.9
Product rule of logarithms
42.10
Quotient rule of logarithms
42.11
Combining product rule and quotient rule in logarithms
42.12
Evaluating logarithms using logarithm rules
42.13
Solving logarithmic equations
42.14
Graphing logarithmic functions
42.15
Finding a logarithmic function given its graph
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