Logarithms as Inverses

Develop logarithm definition, equivalent forms, logarithm graph.

Lesson 2 of 43 parts Practice Flashcards

1/3

Logarithm Definition

Objective: Explain logarithm definition, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b(y)=x ⇔ b^x=y; b>0, b≠1, y>0$

Worked examples

Try itEvaluate log_3(2187).Show answer

Step 1: By definition, log_3(y)=x iff 3^x=y. Step 2: Here 3^7=2187. Step 3: The logarithm equals 7.

Swipe sideways to move between sections

Logarithm Definition

Objective: Explain logarithm definition, apply the rule, and justify relevant restrictions.

Formulas

$log_b(y)=x ⇔ b^x=y; b>0, b≠1, y>0$

Worked examples

Try itEvaluate log_3(2187).Show answer

Step 1: By definition, log_3(y)=x iff 3^x=y. Step 2: Here 3^7=2187. Step 3: The logarithm equals 7.

Equivalent Forms

Objective: Explain equivalent forms, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b(y)=x ⇔ b^x=y; b>0, b≠1, y>0$

Worked examples

Try itEvaluate log_3(2187).Show answer

Step 1: By definition, log_3(y)=x iff 3^x=y. Step 2: Here 3^7=2187. Step 3: The logarithm equals 7.

Logarithm Graph

Objective: Explain logarithm graph, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b(x−h): x>h$

Worked examples

Try itFor f(x)=log_3(x−7), what is the excluded lower boundary of its domain?Show answer

Step 1: A real logarithm requires a positive argument. Step 2: Solve x−7>0 to obtain x>7. Step 3: The boundary 7 is excluded; domain is (7,∞).

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonExponential Growth and Decay Next lessonLogarithm Properties and Equations