Welcome back to Professor Baker's Math Class! In this post, we are diving deep into the world of logarithms. Sections 5-4 and 5-5 are pivotal because they transition us from simply defining what a logarithm is to actually using them as powerful algebraic tools.

Below is a breakdown of the key concepts from the PowerPoint notes and our class discussion. Be sure to review the attached presentation for specific examples and practice problems.

Section 5-4: Properties of Logarithms

Just as exponents have specific laws, logarithms have properties that allow us to manipulate and simplify expressions. These are essential for solving the equations we encounter in the next section.

  • The Product Rule: The logarithm of a product is the sum of the logarithms.
    $$ \log_b(MN) = \log_b(M) + \log_b(N) $$
  • The Quotient Rule: The logarithm of a quotient is the difference of the logarithms.
    $$ \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) $$
  • The Power Rule: This is often the most useful property for moving variables out of the exponent position.
    $$ \log_b(M^p) = p \cdot \log_b(M) $$

We use these properties for two main tasks: Expanding (breaking a complex log into simpler parts) and Condensing (combining multiple logs into a single expression). Mastery of condensing is a prerequisite for solving logarithmic equations.

Section 5-5: Exponential and Logarithmic Equations

Now that we can manipulate the expressions, we apply these skills to solve for $x$. We generally categorize these problems into two types:

1. Solving Exponential Equations

When the variable is in the exponent (e.g., $3^{x+1} = 5$), we have two primary strategies:

  1. Like Bases: If you can express both sides with the same base, you can set the exponents equal to each other.
    $$ b^M = b^N \implies M = N $$
  2. Unlike Bases: If the bases cannot be matched, take the logarithm (usually $\ln$ or $\log$) of both sides and apply the Power Rule to bring the variable down.
    $$ \ln(3^{x+1}) = \ln(5) \implies (x+1)\ln(3) = \ln(5) $$

2. Solving Logarithmic Equations

When the variable is inside the log (e.g., $\log_2(x) + \log_2(x-1) = 3$), follow these steps:

  • Condense: Use the properties from Section 5-4 to combine terms into a single logarithm on one side.
  • Convert: Switch the equation from logarithmic form to exponential form.
    $$ \log_b(x) = y \implies b^y = x $$
  • Solve: Isolate $x$ using standard algebra.

Create a Habit: Check Your Answers!
This is not optional for Section 5-5. The domain of a logarithmic function is $(0, \infty)$. This means you cannot take the log of a negative number or zero. When solving log equations, you may derive extraneous solutions—answers that look algebraically correct but violate the domain rules. Always plug your $x$ back into the original equation to ensure the argument of the logarithm is positive.

Download the PowerPoint below to see these methods applied to various practice problems. Keep practicing, and let's master these logs!