Logarithmic Functions and Their Properties: Study Guide

Logarithmic Functions

Definition and Basic Properties

The logarithmic function to the base a (where a > 0 and a \neq 1) is denoted as \( y = \log_a x \) and is defined by the relationship \( x = a^y \). The domain of the logarithmic function is x > 0, and x is called the argument of the logarithm.

• Common Logarithm: Logarithm with base 10, written as \( \log x \).

• Natural Logarithm: Logarithm with base e (Euler's number), written as \( \ln x \).

Fundamental Properties of Logarithms

Logarithms have several key properties that are essential for simplifying expressions and solving equations:

• Property 1: \( \log_a 1 = 0 \) — The logarithm of 1 to any base is 0.

• Property 2: \( \log_a a = 1 \) — The logarithm of the base to itself is 1.

• Property 3: \( a^{\log_a M} = M \) — Exponential and logarithmic functions are inverses.

• Property 4: \( \log_a a^r = r \) — The logarithm of a base raised to a power equals the exponent.

• Property 5: \( \log_a (M \cdot N) = \log_a M + \log_a N \) — The logarithm of a product is the sum of the logarithms.

• Property 6: \( \log_a \left( \frac{M}{N} \right) = \log_a M - \log_a N \) — The logarithm of a quotient is the difference of the logarithms.

• Property 7: \( \log_a M^r = r \log_a M \) — The logarithm of a power is the exponent times the logarithm.

• Property 8: \( \log_a M = \frac{\log_b M}{\log_b a} \) — Change of base formula, useful for evaluating logarithms with arbitrary bases.

Examples and Applications

• Example 1: Rewrite using the definition of logarithms:

  • \( 2^4 = 16 \) → \( \log_2 16 = 4 \)

  • \( e^x = 6 \) → \( x = \ln 6 \)

  • \( \log_2 5 = b \) → \( 2^b = 5 \)

  • \( \ln 3 = x \) → \( e^x = 3 \)

• Example 2: Find the exact value of the logarithm without a calculator:

  • \( \log_8 8 = 1 \) (since 8 is the base)

  • \( \log_3 25 \) — Not an integer, but can be written as \( \frac{\ln 25}{\ln 3} \) using the change of base formula.

• Example 3: Find the domain of the function:

  • \( f(x) = \ln(x-1) \) — Domain: \( x > 1 \)

  • \( f(x) = \log_2 \left( \frac{5x-1}{3-x} \right) \) — Domain: \( 5x-1 > 0 \) and \( 3-x > 0 \)

Properties of Logarithms: Expanded Use

Logarithmic properties are used to simplify expressions, solve equations, and rewrite logarithmic expressions in different forms.

• Product Rule: \( \log_a (MN) = \log_a M + \log_a N \)

• Quotient Rule: \( \log_a \left( \frac{M}{N} \right) = \log_a M - \log_a N \)

• Power Rule: \( \log_a M^r = r \log_a M \)

• Change of Base: \( \log_a M = \frac{\log_b M}{\log_b a} \)

Solving Logarithmic and Exponential Equations

To solve logarithmic and exponential equations, use the properties above to isolate the variable and rewrite the equation in a solvable form.

• Combine logarithms using product, quotient, and power rules.

• Rewrite logarithmic equations as exponential equations and vice versa.

• Check that solutions are within the domain of the original logarithmic function.

Financial Models: Compound Interest and Continuous Compounding

Logarithmic and exponential functions are used in financial models, such as compound interest and continuous compounding.

• Compound Interest Formula: $A = P \left( 1 + \frac{r}{n} \right)^{nt}$

• Continuous Compounding: $A = P e^{rt}$

• Effective Rate of Interest: $r_e = \left( 1 + \frac{r}{n} \right)^n - 1$ (compounding n times per year), $r_e = e^r - 1$ (continuous compounding)

Exponential Growth and Decay Models

Exponential functions model growth and decay in natural processes, such as population growth, radioactive decay, and cooling.

• Uninhibited Growth/Decay: $N(t) = N_0 e^{kt}$

• Newton's Law of Cooling: $u(t) = T + (u_0 - T) e^{kt}$

Summary Table: Logarithmic Properties

Property	Formula	Description
Identity	$\log_a 1 = 0$	Logarithm of 1 is 0
Base	$\log_a a = 1$	Logarithm of base is 1
Inverse	$a^{\log_a M} = M$	Exponential and logarithmic are inverses
Exponent	$\log_a a^r = r$	Logarithm of base to exponent is exponent
Product	$\log_a (MN) = \log_a M + \log_a N$	Logarithm of product is sum
Quotient	$\log_a \left( \frac{M}{N} \right) = \log_a M - \log_a N$	Logarithm of quotient is difference
Power	$\log_a M^r = r \log_a M$	Logarithm of power is exponent times logarithm
Change of Base	$\log_a M = \frac{\log_b M}{\log_b a}$	Change base of logarithm