Essential Concepts

Logarithmic Properties

Note: These rules only apply to products, quotients, powers, and roots—never to addition or subtraction inside the argument

Change-of-Base Formula

For any positive real numbers [latex]M[/latex], [latex]b[/latex], and [latex]n[/latex] (where [latex]n \neq 1[/latex] and [latex]b \neq 1[/latex]):

• [latex]\log_b(M) = \frac{\ln(M)}{\ln(b)}[/latex] (using natural log)
• [latex]\log_b(M) = \frac{\log(M)}{\log(b)}[/latex] (using common log)

Exponential and Logarithmic Equations

Solving Exponential Equations Using Like Bases

One-to-One Property of Exponential Functions: For any algebraic expressions [latex]S[/latex] and [latex]T[/latex], and any positive real number [latex]b \neq 1[/latex]:

[latex]b^S = b^T[/latex] if and only if [latex]S = T[/latex]

Strategy:

1. Rewrite each side with a common base using exponent properties
2. Apply the one-to-one property to set exponents equal
3. Solve the resulting equation

Solving Exponential Equations Using Logarithms

Property of Logarithmic Equality: If [latex]\log_b(M) = \log_b(N)[/latex], then [latex]M = N[/latex]

When bases cannot be made equal:

1. Apply logarithm to both sides (use [latex]\ln[/latex] if base [latex]e[/latex] is present, [latex]\log[/latex] if base 10 is present, or either otherwise)
2. Use the power rule to bring exponents down
3. Solve for the unknown

Equations with Base e

For [latex]Ae^{kt} = c[/latex]:

1. Isolate exponential: [latex]e^{kt} = \frac{c}{A}[/latex]
2. Apply [latex]\ln[/latex]: [latex]kt = \ln\left(\frac{c}{A}\right)[/latex]
3. Solve for variable

Remember: [latex]\ln(e^x) = x[/latex] and [latex]e^{\ln(x)} = x[/latex] for [latex]x > 0[/latex]

Solving Logarithmic Equations Using the Definition

For [latex]\log_b(S) = c[/latex], convert to exponential form: [latex]b^c = S[/latex]

Strategy:

1. Use properties to write as a single log on one side
2. Convert to exponential form
3. Solve the equation

Solving Logarithmic Equations Using the One-to-One Property

One-to-One Property of Logarithms: For [latex]x > 0[/latex], [latex]S > 0[/latex], [latex]T > 0[/latex], and [latex]b > 0[/latex] (where [latex]b \neq 1[/latex]):

[latex]\log_b(S) = \log_b(T)[/latex] if and only if [latex]S = T[/latex]

Strategy:

1. Use properties to get single logs with same base on each side
2. Set arguments equal
3. Solve the equation
4. Always check for extraneous solutions

Extraneous Solutions

Solutions are extraneous when:

• The logarithm of a negative number or zero would be required
• The solution doesn’t satisfy the original equation

Exponential and Logarithmic Models

Continuous Growth/Decay Model: [latex]A(t) = A_0 e^{kt}[/latex]

Where:

• [latex]A_0[/latex] = initial amount at [latex]t = 0[/latex]
• [latex]k[/latex] = continuous growth rate ([latex]k > 0[/latex] for growth, [latex]k < 0[/latex] for decay)
• [latex]t[/latex] = time
• [latex]A(t)[/latex] = amount at time [latex]t[/latex]

Characteristics:

• Domain: [latex](-\infty, \infty)[/latex]
• Range: [latex](0, \infty)[/latex]
• Horizontal asymptote: [latex]y = 0[/latex]

Half-Life

Time for a quantity to decay to half its original amount:

[latex]t = \frac{\ln(0.5)}{k} = -\frac{\ln(2)}{k}[/latex]

Alternative formula: [latex]A(t) = A_0\left(\frac{1}{2}\right)^{\frac{t}{T}}[/latex] where [latex]T[/latex] is the half-life

Use: Common in radioactive decay and carbon-14 dating

Newton’s Law of Cooling

Temperature of an object in surrounding air:

[latex]T(t) = Ae^{kt} + T_s[/latex]

Where:

• [latex]A[/latex] = difference between initial and surrounding temperature
• [latex]k[/latex] = continuous rate of cooling (negative)
• [latex]T_s[/latex] = surrounding (ambient) temperature
• [latex]t[/latex] = time

Strategy:

1. Set [latex]T_s[/latex] equal to ambient temperature
2. Use initial conditions to find [latex]A[/latex]
3. Use a second data point to find [latex]k[/latex]
4. Use model to make predictions

Logistic Growth Model: [latex]f(x) = \frac{c}{1 + ae^{-bx}}[/latex]

Where:

• [latex]\frac{c}{1+a}[/latex] = initial value
• [latex]c[/latex] = carrying capacity (limiting value/maximum)
• [latex]b[/latex] = constant determined by rate of growth

Choosing the Right Model

Fitting Exponential Models to Data

Using a Graphing Calculator for Regression:

1. Enter data in lists (L1 for input, L2 for output)
2. Create scatter plot to identify pattern
3. Use appropriate regression command:
   • ExpReg for exponential
   • LnReg for logarithmic
   • Logistic for logistic
4. Graph model with data to verify fit
5. Check [latex]r^2[/latex] value (closer to 1 = better fit)

Interpolation vs. Extrapolation

• Interpolation: Predictions within the data range (more reliable)
• Extrapolation: Predictions outside the data range (less reliable, requires careful reasoning)

Key Equations

Product rule for logarithms	[latex]\log_b(MN) = \log_b(M) + \log_b(N)[/latex]
Quotient rule for logarithms	[latex]\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)[/latex]
Power rule for logarithms	[latex]\log_b(M^n) = n\log_b(M)[/latex]
Change-of-base formula	[latex]\log_b(M) = \frac{\log_n(M)}{\log_n(b)} = \frac{\ln(M)}{\ln(b)}[/latex]
One-to-one property (exponential)	[latex]b^S = b^T \Leftrightarrow S = T[/latex]
One-to-one property (logarithmic)	[latex]\log_b(S) = \log_b(T) \Leftrightarrow S = T[/latex]
Continuous growth/decay	[latex]A(t) = A_0 e^{kt}[/latex]
Doubling time	[latex]t = \frac{\ln(2)}{k}[/latex]
Half-life	[latex]t = -\frac{\ln(2)}{k}[/latex] or [latex]A(t) = A_0\left(\frac{1}{2}\right)^{\frac{t}{T}}[/latex]
Newton’s Law of Cooling	[latex]T(t) = Ae^{kt} + T_s[/latex]
Logistic growth	[latex]f(x) = \frac{c}{1 + ae^{-bx}}[/latex]
Exponential regression	[latex]y = ab^x[/latex]
Logarithmic regression	[latex]y = a + b\ln(x)[/latex]

Glossary

carrying capacity

The limiting value [latex]c[/latex] in a logistic model; represents the maximum sustainable population or quantity.

change-of-base formula

A formula that allows evaluation of logarithms with any base using logarithms of another base: [latex]\log_b(M) = \frac{\log_n(M)}{\log_n(b)}[/latex].

concavity

The direction a curve bends; concave up curves bend upward (can hold water), concave down curves bend downward (cannot hold water).

continuous growth/decay model

A model of the form [latex]A(t) = A_0 e^{kt}[/latex] where [latex]k > 0[/latex] represents growth and [latex]k < 0[/latex] represents decay.

extraneous solution

A solution that emerges algebraically but doesn’t satisfy the original equation; common when logarithms of negative numbers or zero would be required.

extrapolation

Using a model to make predictions outside the range of original data; less reliable and requires careful reasoning.

half-life

The time required for an exponentially decaying quantity to reduce to half its original amount; calculated as [latex]t = -\frac{\ln(2)}{k}[/latex].

identity property of logarithms

The logarithm of the base to itself equals 1: [latex]\log_b(b) = 1[/latex].

interpolation

Using a model to make predictions within the range of original data; generally more reliable than extrapolation.

inverse property of logarithms

Logarithms and exponentials undo each other: [latex]b^{\log_b(x)} = x[/latex] and [latex]\log_b(b^x) = x[/latex].

logistic growth

Growth that is exponential at first but slows as it approaches a maximum value (carrying capacity); modeled by [latex]f(x) = \frac{c}{1 + ae^{-bx}}[/latex].

Newton’s Law of Cooling

A model describing how an object’s temperature changes over time to equalize with surrounding temperature: [latex]T(t) = Ae^{kt} + T_s[/latex].

one-to-one property of exponential functions

If [latex]b^S = b^T[/latex], then [latex]S = T[/latex] for any positive real number [latex]b \neq 1[/latex].

one-to-one property of logarithmic functions

If [latex]\log_b(S) = \log_b(T)[/latex], then [latex]S = T[/latex] for positive real numbers [latex]S[/latex], [latex]T[/latex], and base [latex]b > 0[/latex], [latex]b \neq 1[/latex].

power rule for logarithms

The logarithm of a power equals the exponent times the logarithm: [latex]\log_b(M^n) = n\log_b(M)[/latex].

product rule for logarithms

The logarithm of a product equals the sum of the logarithms: [latex]\log_b(MN) = \log_b(M) + \log_b(N)[/latex].

property of logarithmic equality

If [latex]\log_b(M) = \log_b(N)[/latex], then [latex]M = N[/latex] for any positive real numbers [latex]M[/latex], [latex]N[/latex], and base [latex]b > 0[/latex], [latex]b \neq 1[/latex].

quotient rule for logarithms

The logarithm of a quotient equals the difference of the logarithms: [latex]\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)[/latex].

radiocarbon dating

A method for determining the age of organic materials by measuring the ratio of carbon-14 to carbon-12, based on carbon-14’s half-life of 5,730 years.

regression

A method of fitting an algebraic model to data

zero property of logarithms

The logarithm of 1 to any base equals 0: [latex]\log_b(1) = 0[/latex].