Lesson 5.2 Logaritgmic Functions; Properties of Logarithms

Key Objectives

  • Understand how logarithmic functions work and how they relate to exponential functions.

  • Learn how to change equations between logarithmic and exponential forms.

  • Know how to use common logarithms and natural logarithms.

  • Practice using properties of logarithms.

Logarithmic Functions and Properties of Logarithms

Overview Logarithmic functions help us understand and rewrite exponential equations. They are really useful in real life, like in predicting how many people in the U.S. might have diabetes in the future.

Example: Diabetes Projections

From 2010 to 2050, it is believed that a third of U.S. adults will have diabetes. We can use a logarithmic equation to model this:

  • p(x) = -12.975 + 11.851 ln(x), where x is the number of years after 2000. To find out when this number is 33%, we can graph two equations:

  • y₁ = -12.975 + 11.851 ln x

  • y = 33 The point where they meet tells us our answer.

What are Logarithmic Functions?

Logarithmic functions are like the opposites of exponential functions. If we say y = f(x), to find the inverse function, we switch x and y and solve for y. For any exponential function like y = b^x, the inverse is a logarithmic function.

Definition of Logarithmic Function

A logarithmic function is written as:

  • y = log_b(x) where x must be more than 0, b must be more than 0, and b cannot be 1. This means that log_b(x) is the power we need to raise b to get x.

Converting Forms

Exponential to Logarithmic Form

To change from exponential to logarithmic form:

  • y = log_b(x) is the same as x = b^y.

  • Examples:

    • If 3² = 9, then log₃(9) = 2.

    • If 4^{-1} = ¼, then log₄(¼) = -1.

Logarithmic to Exponential Form

To change from logarithmic to exponential form:

  • Examples:

    • log₂(16) = 4 means 2⁴ = 16.

    • log₁₀(0.0001) = -4 means 10^{-4} = 0.0001.

Evaluating Logarithms

For example, if we have y = log₂(x), we change it into exponential form:

  • Then 2^y = x.

  • For x = 8, that means y = 3 since 2³ = 8, so log₂(8) is 3.

  • Another example: log₄(1/16) is -2.

Graphing Logarithmic Functions

To graph y = log₂(x), we can find the x-values for different y-values. Common traits of these graphs:

  • They go up if the base is more than 1.

  • They go down if the base is between 0 and 1.

Common Logarithms and Natural Logarithms

  • Base 10 logarithms (written as log x) are known as common logarithms and are useful because we often use base 10 in everything.

  • Natural logarithms use the special number e (about 2.718), written as y = ln(x).

Example: pH Scale

To measure how acidic something is, we use a pH scale:

  • pH = -log[H⁺], where lower numbers mean more acidic and higher numbers mean more basic.

Transformations of Logarithmic Functions

You can shift logarithmic graphs left or right and make them taller or shorter:

  • y = log(x + 3) moves it 3 units left;

  • y = 4 + log(x - 2) moves it 2 units right and up 4 units;

  • y = (1/2) log(x) makes it shorter.

Additional Logarithmic Properties

  • Product Property: log_b(MN) = log_b(M) + log_b(N)

  • Quotient Property: log_b(M/N) = log_b(M) - log_b(N)

  • Power Property: log_b(M^k) = k * log_b(M)

Rewriting Logarithmic Expressions

We can simplify logarithmic expressions using the properties:

  • Example: log₃(x) + 4 log₃(y) becomes log₃(xy^4).

  • Example: ln(5x) - 3 ln(z) becomes ln(5x/z³).

Richter Scale

The strength of earthquakes is measured by a logarithmic formula: R = log(I/I₀), where I₀ is a set level of intensity. Each number increase on the Richter scale means an increase in intensity by 10 times!