4.5 Logarithmic Properties

Logarithmic Properties and Applications

Logarithmic properties give you a toolkit for breaking apart and recombining log expressions. Once you know the three core rules, you can simplify complex expressions, solve equations that would otherwise be impossible, and convert between bases on any calculator.

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Rules for logarithmic simplification

Three properties do most of the heavy lifting. Each one connects a log operation to a simpler arithmetic operation.

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

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

For example: $\log_2(6x) = \log_2(6) + \log_2(x)$. You can break $\log_2(6)$ down further since $6 = 2 \cdot 3$: $\log_2(2) + \log_2(3) + \log_2(x) = 1 + \log_2(3) + \log_2(x)$.

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

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

For example: $\log_5\left(\frac{25}{x}\right) = \log_5(25) - \log_5(x) = 2 - \log_5(x)$.

Power Rule: The log of something raised to a power lets you pull the exponent out front as a multiplier.

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

For example: $\log_3(x^4) = 4\log_3(x)$.

Why do these work? Logarithms are exponents. The product rule mirrors the exponent rule $b^m \cdot b^n = b^{m+n}$, the quotient rule mirrors $\frac{b^m}{b^n} = b^{m-n}$, and the power rule mirrors $(b^m)^n = b^{mn}$. Keeping that connection in mind makes the properties feel less arbitrary.

Expansion and condensing of logarithmic expressions

Expanding means using the properties to break a single log into multiple simpler logs. Condensing means combining multiple logs into one. These are inverse processes, and you'll need both directions.

Expanding example: Expand $\log_2(8x^3)$.

1. Apply the product rule: $\log_2(8) + \log_2(x^3)$
2. Apply the power rule to the second term: $\log_2(8) + 3\log_2(x)$
3. Evaluate $\log_2(8) = 3$ since $2^3 = 8$: $3 + 3\log_2(x)$

Expanding example with a quotient: Expand $\log_b\left(\frac{x^2}{y}\right)$.

1. Apply the quotient rule: $\log_b(x^2) - \log_b(y)$

2. Apply the power rule: $2\log_b(x) - \log_b(y)$

Condensing example: Condense $\log_3(x) + \log_3(y)$.

• Apply the product rule in reverse: $\log_3(xy)$

Condensing example with the power rule: Condense $2\log_4(x)$.

• Apply the power rule in reverse: $\log_4(x^2)$

A common mistake when condensing: you can only combine logs that have the same base. $\log_2(x) + \log_3(y)$ cannot be condensed into a single logarithm.

Change-of-base formula for logarithms

Most calculators only have two log buttons: $\log$ (base 10) and $\ln$ (base $e$). The change-of-base formula lets you evaluate a log of any base using either of these:

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

You can choose any new base $a$, but in practice you'll use base 10 or base $e$:

• $\log_5(x) = \frac{\log(x)}{\log(5)}$ or equivalently $\frac{\ln(x)}{\ln(5)}$

Evaluating example: Find $\log_3(5)$.

1. Apply the formula: $\log_3(5) = \frac{\log(5)}{\log(3)}$
2. Use a calculator: $\frac{0.69897}{0.47712} \approx 1.465$

Both $\log$ and $\ln$ give the same final answer here. Pick whichever you prefer.

Logarithmic Equations and Functions

Solving logarithmic equations

The general strategy is to isolate the log, convert to exponential form, then solve.

Step-by-step process:

1. Isolate the logarithmic term using algebra. If you have multiple log terms with the same base, condense them into one log first.
2. Convert to exponential form using the definition: $\log_b(x) = y \Leftrightarrow b^y = x$.
3. Solve the resulting equation.
4. Check for extraneous solutions. Plug your answer back into the original equation. The argument of every logarithm must be positive. If a solution makes any log argument zero or negative, throw it out.

Example: Solve $2\log_3(x) + 1 = 7$.

1. Isolate: subtract 1, then divide by 2: $\log_3(x) = 3$
2. Convert: $3^3 = x$
3. Solve: $x = 27$
4. Check: $2\log_3(27) + 1 = 2(3) + 1 = 7$ ✓

Step 4 matters most when you've squared something or when the original equation has logs of expressions like $(x - 5)$, where negative $x$-values could sneak in.

Graphing of logarithmic functions

The function $f(x) = \log_b(x)$ is the inverse of the exponential function $g(x) = b^x$. That inverse relationship controls everything about the graph.

• Domain: $(0, \infty)$ — you can only take the log of positive numbers
• Range: $(-\infty, \infty)$
• Vertical asymptote at $x = 0$ (the y-axis). The curve approaches it but never touches it.
• Key point: The graph always passes through $(1, 0)$ because $\log_b(1) = 0$ for any valid base.
• Direction: Increasing when $b > 1$, decreasing when $0 < b < 1$.

Since $f(x) = \log_b(x)$ and $g(x) = b^x$ are inverses, their graphs are reflections of each other over the line $y = x$. This also gives you two identities that show up frequently:

$\log_b(b^x) = x \quad \text{and} \quad b^{\log_b(x)} = x$

These are useful for simplifying expressions. If you see $\log_5(5^{3x})$, it collapses straight to $3x$.

Real-world applications of logarithms

Logarithms appear whenever you need to "undo" an exponential or work with quantities that span huge ranges.

Compound Interest: The formula $A = P\left(1 + \frac{r}{n}\right)^{nt}$ gives the future value of an investment. To solve for time $t$, you take the log of both sides:

$t = \frac{\log(A/P)}{n \cdot \log\left(1 + \frac{r}{n}\right)}$

For example, to find how long it takes to double an investment at 6% compounded monthly, set $A/P = 2$, $r = 0.06$, $n = 12$, and solve: $t = \frac{\log(2)}{12 \cdot \log(1.005)} \approx 11.58$ years.

Richter Scale: Earthquake magnitude is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the seismic wave amplitude and $A_0$ is a reference amplitude. Because it's logarithmic, each whole number increase means 10 times the amplitude. A magnitude 5 earthquake has seismic waves with 10 times the amplitude of a magnitude 4.

Other logarithmic scales:

• pH scale (chemistry): $\text{pH} = -\log[H^+]$, measuring acidity
• Decibel scale (physics): measures sound intensity logarithmically
• Exponential growth/decay: taking logs of both sides lets you solve for rates or time in population models and radioactive decay (half-life calculations)