Logarithms as Inverses

Explore logarithm definition, equivalent forms, logarithm graph.

Lesson 3 of 53 parts Practice Flashcards

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Logarithm Definition

Objective: Explain logarithm definition and use it to solve or interpret a problem.

For base b>0, b≠1, log_b(y)=x means b^x=y with y>0. A logarithm reports an exponent, not a multiplication. No, real logarithm arguments must be positive.

Worked examples

Try itLogarithms as Inverses — practice 1.1: Evaluate log_3(243).Show answer

Step 1: By definition, log_3(y)=x iff 3^x=y. Step 2: Here 3^5=243. Step 3: The logarithm equals 5.

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Logarithm Definition

Objective: Explain logarithm definition and use it to solve or interpret a problem.

For base b>0, b≠1, log_b(y)=x means b^x=y with y>0. A logarithm reports an exponent, not a multiplication. No, real logarithm arguments must be positive.

Worked examples

Try itLogarithms as Inverses — practice 1.1: Evaluate log_3(243).Show answer

Step 1: By definition, log_3(y)=x iff 3^x=y. Step 2: Here 3^5=243. Step 3: The logarithm equals 5.

Equivalent Forms

Objective: Explain equivalent forms and use it to solve or interpret a problem.

Convert a logarithmic equation to exponential form to evaluate or solve it, then respect positivity of its argument and allowed base. Positive bases other than one.

Worked examples

Try itLogarithms as Inverses — practice 2.1: Evaluate log_4(65536).Show answer

Step 1: By definition, log_4(y)=x iff 4^x=y. Step 2: Here 4^8=65536. Step 3: The logarithm equals 8.

Logarithm Graph

Objective: Explain logarithm graph and use it to solve or interpret a problem.

For b>1, log_b(x) increases on x>0 and has a vertical asymptote at x=0; it is the inverse graph of y=b^x. The logarithm input must be positive.

Worked examples

Try itLogarithms as Inverses — practice 3.1: For f(x)=1/(x−11), which x-value is excluded?Show answer

Step 1: The denominator cannot be zero. Step 2: Solve x−11=0. Step 3: Therefore x=11 is excluded.

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