6.5 Logarithmic Properties

Logarithmic properties give you a set of rules for breaking apart, combining, and rewriting expressions that involve logs. These properties are essential for solving logarithmic and exponential equations throughout College Algebra, and they show up again in precalculus and calculus. This section covers the core rules, how to expand and condense expressions, the change-of-base formula, solving equations, and key features of logarithmic graphs.

Logarithmic Properties

Rules for logarithmic simplification

Three core rules let you rewrite logarithmic expressions. Each one connects a log operation to a simpler arithmetic operation.

Product Rule: The log of a product equals the sum of the logs.

$\log_b(M \cdot N) = \log_b(M) + \log_b(N)$

• Example: $\log_2(6 \cdot 8) = \log_2(6) + \log_2(8)$

Quotient Rule: The log of a quotient equals the difference of the logs.

$\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Example: $\log_5\left(\frac{25}{5}\right) = \log_5(25) - \log_5(5) = 2 - 1 = 1$

Power Rule: The log of a value raised to a power equals the exponent times the log.

$\log_b(M^n) = n \cdot \log_b(M)$

• Example: $\log_3(9^2) = 2 \cdot \log_3(9) = 2 \cdot 2 = 4$

All three rules require the same base $b$ on every log in the expression. You can't combine logs with different bases using these rules.

Expansion and condensation of logarithms

Expanding means taking a single log and breaking it into multiple simpler logs using the three rules above. Work from the outside in: handle quotients and products first, then bring down exponents.

• Example: Expand $\log_2(3x^2y)$
  1. Apply the product rule to split the three factors: $\log_2(3) + \log_2(x^2) + \log_2(y)$
  2. Apply the power rule to $\log_2(x^2)$: $\log_2(3) + 2\log_2(x) + \log_2(y)$

Condensing is the reverse: you combine multiple logs into a single log. Move coefficients up as exponents first (power rule in reverse), then combine using the product and quotient rules.

• Example: Condense $\log_4(2) + \log_4(x) - 3\log_4(y)$
  1. Apply the power rule in reverse: $\log_4(2) + \log_4(x) - \log_4(y^3)$
  2. Combine the sum using the product rule: $\log_4(2x) - \log_4(y^3)$
  3. Combine the difference using the quotient rule: $\log_4\left(\frac{2x}{y^3}\right)$

A common mistake is trying to condense logs that are being multiplied together, like $\log_4(2) \cdot \log_4(x)$. The product rule only applies when logs are being added, not multiplied.

Change-of-base formula for logarithms

Most calculators only have buttons for common log (base 10) and natural log (base $e$). The change-of-base formula lets you evaluate a log with any base by converting it:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

You can pick any new base $a$, but base 10 or base $e$ are the practical choices since your calculator handles those directly.

Steps to apply the formula:

1. Identify the log you need to convert (e.g., $\log_3(5)$)
2. Pick a convenient new base (base 10 or $e$)
3. Place the log of the argument in the numerator and the log of the old base in the denominator
4. Evaluate with a calculator

• Example: $\log_3(5) = \frac{\log(5)}{\log(3)} = \frac{0.6990}{0.4771} \approx 1.465$

Solving logarithmic equations

The general strategy is to isolate the log, convert to exponential form, solve, and then check your answer.

Example: Solve $\log_2(3x - 1) = 4$

1. Isolate the logarithm. It's already isolated here: $\log_2(3x - 1) = 4$

2. Convert to exponential form using the definition (if $\log_b(x) = y$, then $b^y = x$): $2^4 = 3x - 1$

3. Solve for the variable: $16 = 3x - 1 \rightarrow 17 = 3x \rightarrow x = \frac{17}{3}$

4. Check the solution. Plug back in: $3\left(\frac{17}{3}\right) - 1 = 16$, and $\log_2(16) = 4$. Also confirm the argument is positive: $16 > 0$. ✓

Always check that your solution keeps the argument of every log positive. If a solution makes the argument zero or negative, it's extraneous and you must throw it out. This is the most common place students lose points.

Real-world applications of logarithms

Logarithmic scales are used whenever quantities span a huge range of values. Instead of comparing raw numbers, a log scale compresses them into manageable units.

• Richter scale (earthquakes): Each whole-number increase represents a tenfold increase in measured amplitude. A magnitude 6 earthquake is 10 times stronger than a magnitude 5.
• Decibel scale (sound): Each 10-decibel increase represents a tenfold increase in sound intensity. A 90 dB sound is 10 times more intense than an 80 dB sound.
• pH scale (chemistry): Each unit decrease in pH represents a tenfold increase in hydrogen ion concentration. A solution with pH 3 is 10 times more acidic than one with pH 4.
• Exponential growth and decay: Logarithms help you solve for time in models like population growth, radioactive decay, and compound interest (e.g., "How long until my investment doubles?").

Properties of logarithmic graphs

The parent function $y = \log_b(x)$ has several key features:

• Domain: $(0, \infty)$. You can only take the log of a positive number.
• Range: $(-\infty, \infty)$. The output can be any real number.
• Vertical asymptote at $x = 0$. The graph gets closer and closer to the y-axis but never touches it.
• x-intercept at $(1, 0)$, because $\log_b(1) = 0$ for any base.
• Direction: If $b > 1$, the graph increases (rises left to right). If $0 < b < 1$, the graph decreases.

Transformations follow the same patterns as other functions:

• $\log_b(x) + k$ shifts the graph up by $k$ units
• $\log_b(x - h)$ shifts the graph right by $h$ units (and moves the asymptote to $x = h$)
• $a \cdot \log_b(x)$ stretches vertically by a factor of $|a|$ (compresses if $0 < |a| < 1$)
• $-\log_b(x)$ reflects the graph across the x-axis

Watch the horizontal shift carefully: $\log_b(x - h)$ moves the vertical asymptote from $x = 0$ to $x = h$. This also changes the domain to $(h, \infty)$.

Logarithms and inverse functions

Logarithmic functions and exponential functions are inverses of each other. This means:

• If $f(x) = b^x$, then $f^{-1}(x) = \log_b(x)$
• The domain of the log function is the range of the exponential function, and vice versa
• Graphically, $y = \log_b(x)$ is the reflection of $y = b^x$ over the line $y = x$

This inverse relationship also gives you two identities that are useful for simplifying:

• $\log_b(b^x) = x$
• $b^{\log_b(x)} = x$

These say that a log and an exponential with the same base "undo" each other, which is exactly what inverse functions do.