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Excursion: Logarithmic Functions to Other Bases

Learn how to evaluate logarithms to any base with and without a calculator, solve exponential and logarithmic equations to any base, and simplify and evaluate logarithmic expressions to any base. Examples and step-by-step explanations provided.

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Excursion: Logarithmic Functions to Other Bases

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  1. Excursion: LogarithmicFunctions to Other Bases 5.5.A Objectives: Evaluate logarithms to any base with and without a calculator. Solve Exponential and logarithmic equations to any base by using an equivalent equation. Use properties and laws of logarithms to simplify and evaluate logarithmic expressions to any base.

  2. Example #1 • Without using a calculator, find each value. B. A. C.

  3. Example #2 • Solve each equation for x.

  4. Example #3 • Solve the equation.

  5. Example #4 • Simplify and write each expression as a single logarithm.

  6. Example #4 • Simplify and write each expression as a single logarithm. Use the hint below just as was done with bases of e and 10.

  7. Example #4 • Simplify and write each expression as a single logarithm. And again an alternative approach:

  8. Change of Base Formula • This formula is the single reason why calculators only build in the base 10 (common log) and base e (natural log). With it, it allows a logarithm of any base to be evaluated. A good way to remember which number goes on top is the fact that the base of the original logarithm always goes in the bottom.

  9. Example #5 • Evaluate the following logarithm using the change of base formula. On the first problem the base was 5, so log 5 was placed in the bottom. On the second problem, ¼ was the base, so log ¼ was placed in the bottom. Also remember either the common log or natural log works on the calculator, just don’t mix and match them on the same ratio.

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