Important Logarithmic Functions

Why This Matters

Logarithmic functions are the mathematical backbone of Unit 2 in AP Precalculus, and they're not going away—you'll see them in modeling questions, equation-solving problems, and function analysis throughout the exam. The key insight is that logarithms are inverses of exponential functions, which means every property of exponentials has a mirror property in logarithms. You're being tested on your ability to move fluidly between these two representations, apply transformation rules, and recognize when a real-world scenario calls for logarithmic modeling.

What makes logarithms tricky is that they compress huge ranges of values into manageable numbers—that's why they show up in pH scales, earthquake measurements, and sound intensity. The College Board wants you to understand not just how to manipulate logarithmic expressions, but why these functions behave the way they do: their restricted domains, their characteristic shapes, and their inverse relationship with exponentials. Don't just memorize formulas—know what concept each property illustrates and when to deploy it.

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The Two Base Functions You Must Know

Every logarithmic calculation ultimately traces back to one of two special bases. These are the functions your calculator handles directly, and they form the foundation for all logarithmic work.

Natural Logarithm (ln x)

• Base $e \approx 2.718$—this irrational number appears naturally in continuous growth and decay models
• Inverse relationship: $\ln(e^x) = x$ and $e^{\ln x} = x$, which you'll use constantly when solving equations
• Domain restriction: only defined for $x > 0$, reflecting that you can only take logarithms of positive numbers

Common Logarithm ($\log_{10} x$)

• Base 10—written simply as "log" when no base is specified, matching our decimal number system
• Inverse relationship: $\log_{10}(10^x) = x$ and $10^{\log_{10} x} = x$, essential for order-of-magnitude calculations
• Calculator default: most scientific calculators have a dedicated LOG button for this function

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Properties That Transform Expressions

These three rules let you break apart or combine logarithmic expressions—they're the algebraic workhorses of logarithm manipulation.

Product Rule

• $\log_b(xy) = \log_b(x) + \log_b(y)$—multiplication inside becomes addition outside
• Expanding expressions: use this to break a single logarithm into simpler parts for solving
• Condensing expressions: reverse it to combine multiple logarithms into one, often needed before exponentiating

Quotient Rule

• $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$—division inside becomes subtraction outside
• Mirror of product rule: these two rules together handle all multiplication and division inside logarithms
• Common error alert: students often confuse this with $\frac{\log_b(x)}{\log_b(y)}$, which is completely different

Power Rule

• $\log_b(x^k) = k \cdot \log_b(x)$—exponents inside become coefficients outside
• Most powerful for solving: this rule lets you "bring down" unknown exponents to solve exponential equations
• Works with any exponent: $k$ can be negative, fractional, or any real number

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The Change of Base Formula

This single formula connects all logarithmic bases and makes any logarithm calculator-friendly.

Change of Base Formula

• $\log_b(a) = \frac{\log_k(a)}{\log_k(b)}$—converts any base $b$ logarithm to a ratio of base $k$ logarithms
• Calculator application: since calculators only have ln and log₁₀, use $\log_5(20) = \frac{\ln(20)}{\ln(5)}$ or $\frac{\log(20)}{\log(5)}$
• Equation solving: essential when comparing or combining logarithms with different bases in the same problem

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Inverse Function Relationships

The exponential-logarithmic inverse relationship is the conceptual heart of this unit—every graphing and solving technique flows from it.

Exponential-Logarithmic Inverses

• $f(x) = b^x$ and $g(x) = \log_b(x)$ are inverses—meaning $f(g(x)) = x$ and $g(f(x)) = x$ for appropriate domains
• Graphical reflection: inverse functions reflect across the line $y = x$, so if you know one graph, you know the other
• Domain-range swap: exponential has domain $(-\infty, \infty)$ and range $(0, \infty)$; logarithm flips these exactly

Solving Strategy: Converting Forms

• Exponential to logarithmic: $b^y = x$ becomes $\log_b(x) = y$—use when the variable is in the exponent
• Logarithmic to exponential: $\log_b(x) = y$ becomes $b^y = x$—use when you need to isolate the argument
• Check for extraneous solutions: always verify answers don't produce logarithms of negative numbers or zero

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Graphing and Transformations

Understanding the parent function's shape lets you quickly sketch any transformed logarithm.

Parent Function Behavior

• Key point $(1, 0)$—every logarithmic function passes through this point because $\log_b(1) = 0$ for any base
• Vertical asymptote at $x = 0$—the function approaches $-\infty$ as $x$ approaches 0 from the right
• Increasing for $b > 1$: larger inputs produce larger outputs, but the rate of increase slows (logarithmic growth)

Domain and Range

• Domain: $(0, \infty)$—you cannot take the logarithm of zero or negative numbers
• Range: $(-\infty, \infty)$—logarithmic outputs can be any real number, positive or negative
• Transformation effects: horizontal shifts change the asymptote location; vertical shifts change the horizontal intercept

Transformations

• $y = \log_b(x - h) + k$—shifts right by $h$ and up by $k$, moving the asymptote to $x = h$
• $y = a\log_b(x)$—vertical stretch by factor $|a|$; if $a < 0$, reflects across the x-axis
• $y = \log_b(-x)$—reflects across the y-axis, creating domain $(-\infty, 0)$ with asymptote at $x = 0$

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Real-World Applications

Logarithms appear whenever quantities span multiple orders of magnitude or involve proportional change.

pH Scale

• $\text{pH} = -\log_{10}[\text{H}^+]$—measures hydrogen ion concentration, where lower pH means higher acidity
• Logarithmic compression: each pH unit represents a 10-fold change in concentration
• Negative sign: ensures that higher concentrations (more acidic) give lower pH values

Richter Scale

• Logarithmic earthquake measurement—each whole number increase represents 10 times more ground motion
• Energy scaling: the energy released increases by about 31.6 times per unit on the scale
• Why logarithmic: earthquakes vary by factors of millions; a linear scale would be impractical

Decibel Scale

• $\text{dB} = 10\log_{10}\left(\frac{I}{I_0}\right)$—measures sound intensity relative to a reference level $I_0$
• Doubling intensity: adds about 3 dB (since $10\log_{10}(2) \approx 3$)
• Human perception: our ears respond logarithmically to sound, making this scale psychologically meaningful

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Quick Reference Table

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Self-Check Questions

1. Which two logarithmic properties would you combine to simplify $\log_3(x^2y) - \log_3(z)$ into a single logarithm?

2. If $\log_b(x) = 2$ and $\log_b(y) = 5$, what is $\log_b\left(\frac{x^3}{y}\right)$? Which properties did you use?

3. Compare and contrast the graphs of $y = \ln(x)$ and $y = \ln(x - 2)$. What changes and what stays the same?

4. An FRQ gives you $5^{2x+1} = 17$ and asks for an exact answer. Write the solution in terms of logarithms and explain why you chose your approach.

5. The pH of a solution drops from 6 to 4. By what factor did the hydrogen ion concentration increase? Explain using the definition of pH.