2.9 Logarithmic Functions

Introduction to Logarithmic Expressions

  • Welcome and overview of the topic

  • Presenter: Mr. Sullivan (Sully)

Definition of Logarithms

  • Logarithm Concept: log base B of C equals A if and only if B to the exponent A equals C.

  • Example:

    • log base 4 of 16 = 2

    • Calculation: 4^2 = 16

    • Interpretation: How many times must we multiply the base (4) to get to C (16)? Here, it’s 2 times.

Constraints on Logarithms

  • Negative Logarithms: Not allowed; log cannot have a negative number.

  • Negative Bases: Not allowed in logarithmic expressions.

  • Base Characteristics:

    • B must be greater than 0 and cannot equal 1.

Rewriting Logarithmic Expressions as Exponents

  • Example:

    • log base 3 of 81 = 4

    • Rewrite as: 3^4 = 81 (how many of base 3 reaches 81? Four of them.)

  • Example (Fractional Exponents):

    • log base 16 of 4

    • Rewrite as: 16^(1/2) = 4 (square root of 16 is 4).

Rewriting Exponents as Logarithms

  • Example:

    • log base 125 of 5 = 1/3

    • Calculation: 125^(1/3) = 5 (third root of 125 gives 5).

  • Common Logarithm:

    • log base 10 of 1000 = 3 becomes simply log 1000 = 3

    • Common log does not need the base written (assumed to be 10).

Finding Values of Logarithms

  • Example Calculation: To find log 1000:

    • 10^x = 100, where x = 2.

  • Positive and Negative Exponents:

    • Example with fractions: 2^Y = 1/32 → Y = -5 (negative due to the fraction).

Utilization of Calculators for Logarithms

  • Example Calculation: Using calculator functions to find log base 4 of 25 = 2.322.

  • Common Logarithm Button:

    • Direct function for base 10 with calculator.

  • Change of Base Formula: If calculators lack a log base function, use change of base to find log values.

Linear vs. Logarithmic Scales

  • Linear Scale: Each unit increases by a constant rate.

    • Example: Values such as 2000, 4000, etc., which appear spread out and show a large outlier.

  • Logarithmic Scale: Each unit increases by a power of 10.

    • Example: Values would appear more uniformly spaced when plotted as logs.

Practical Application and Example of Logarithmic Scale

  • Plotting logs based on video lengths (e.g., average length of Sully’s videos and others).

  • Comparison of distributions on linear versus logarithmic scales.

Hands-On Activity

  • Find the log base 5 of specific values and plot them.

Conclusion

  • Brief final review of logarithmic properties and reminder of importance in calculations.

  • Encouragement to practice and understand logarithmic expressions better.