Logarithmic Functions

Rick Says Welcome!

Logarithms are just exponentials looking in a mirror. Reflect on that for a moment! In the last chapter, we asked "What do I get when I raise \(2\) to the power \(x\)?" Now we flip the question: "What power do I need to raise \(2\) to in order to get a certain number?" That's what logarithms answer. Every input has its output — and logarithms help us find the input when we only know the output!

Summary

This chapter introduces logarithmic functions as the inverses of exponential functions. Students explore the log-exponential inverse relationship, then study common logarithms (\(\log_{10}\)) and natural logarithms (\(\ln\)). Key skills include applying logarithm laws, using the change of base rule, and solving logarithmic equations. The chapter also covers logarithmic scales and graphing logarithmic functions.

Concepts Covered

This chapter covers the following 9 concepts from the learning graph:

1. Log-Exponential Inverse
2. Logarithmic Function
3. Common Logarithm
4. Natural Logarithm
5. Logarithm Laws
6. Change of Base Rule
7. Solving Log Equations
8. Logarithmic Scale
9. Graphing Logarithms

Prerequisites

This chapter builds on concepts from:

• Chapter 6: Inverse and Composite Functions
• Chapter 7: Exponential Functions

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The Log-Exponential Inverse Relationship

In Chapter 6, we learned that inverse functions "undo" each other. In Chapter 7, we met exponential functions but couldn't solve equations like \(2^x = 5\) algebraically. Now we meet the tool that fills that gap.

The log-exponential inverse relationship is the fundamental connection:

\[\text{If } a^y = x, \quad \text{then } y = \log_a x\]

In words: the logarithm base \(a\) of \(x\) answers the question "What power do I raise \(a\) to in order to get \(x\)?"

This means exponential and logarithmic functions are inverses:

\[f(x) = a^x \quad \text{and} \quad f^{-1}(x) = \log_a x\]

The key conversion between the two forms:

\[a^y = x \iff y = \log_a x\]

Worked Example 1: Convert between exponential and logarithmic form.

Exponential Form	Logarithmic Form
\(2^3 = 8\)	\(\log_2 8 = 3\)
\(10^2 = 100\)	\(\log_{10} 100 = 2\)
\(5^{-1} = 0.2\)	\(\log_5 0.2 = -1\)
\(e^0 = 1\)	\(\ln 1 = 0\)

Try it yourself: Write \(3^4 = 81\) in logarithmic form, and write \(\log_2 32 = 5\) in exponential form.

Click to reveal answer

\(\log_3 81 = 4\) and \(2^5 = 32\).

The Logarithmic Function

A logarithmic function has the form:

\[f(x) = \log_a x, \quad a > 0, \; a \neq 1\]

Key properties:

• Domain: \((0, \infty)\) — you can only take the logarithm of a positive number
• Range: \((-\infty, \infty)\) — logarithms can output any real number
• \(x\)-intercept: \((1, 0)\) since \(\log_a 1 = 0\) for any base
• Vertical asymptote: \(x = 0\) (the \(y\)-axis)
• The graph is a reflection of \(y = a^x\) over the line \(y = x\)

Notice how the domain and range of \(\log_a x\) are the reverse of \(a^x\). This makes sense — they're inverse functions, so inputs and outputs are swapped.

Common Logarithm

The common logarithm uses base \(10\) and is written simply as \(\log x\) (without a subscript):

\[\log x = \log_{10} x\]

Common logarithms are called "common" because base \(10\) matches our decimal number system. Your calculator's LOG button gives common logarithms.

Useful values to know:

• \(\log 1 = 0\) (since \(10^0 = 1\))
• \(\log 10 = 1\) (since \(10^1 = 10\))
• \(\log 100 = 2\) (since \(10^2 = 100\))
• \(\log 1000 = 3\) (since \(10^3 = 1000\))

Natural Logarithm

The natural logarithm uses base \(e\) and is written \(\ln x\):

\[\ln x = \log_e x\]

Natural logarithms pair with the natural exponential \(e^x\) as inverses:

\[e^{\ln x} = x \quad \text{and} \quad \ln(e^x) = x\]

Your calculator's LN button gives natural logarithms.

Rick's Key Insight

Common logs answer: "How many digits does this number have?" (roughly). Natural logs answer: "How many times does \(e\) need to multiply itself?" Common logs are great for orders of magnitude. Natural logs are what nature actually uses. Both are useful — but in the IB, \(\ln\) appears much more often.

Logarithm Laws

The logarithm laws are rules that simplify logarithmic expressions. They follow directly from the laws of exponents.

Law 1 — Product Rule:

\[\log_a(mn) = \log_a m + \log_a n\]

The log of a product equals the sum of the logs.

Law 2 — Quotient Rule:

\[\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n\]

The log of a quotient equals the difference of the logs.

Law 3 — Power Rule:

\[\log_a(m^k) = k \cdot \log_a m\]

The log of a power brings the exponent out front as a multiplier.

These three laws, together with the identity \(\log_a a = 1\) and \(\log_a 1 = 0\), allow you to manipulate virtually any logarithmic expression.

Worked Example 2: Simplify \(\log_2 8 + \log_2 4\).

\[\log_2 8 + \log_2 4 = \log_2(8 \times 4) = \log_2 32 = 5\]

Worked Example 3: Express \(\log_a\left(\frac{x^3\sqrt{y}}{z^2}\right)\) in terms of \(\log_a x\), \(\log_a y\), and \(\log_a z\).

\[= \log_a x^3 + \log_a \sqrt{y} - \log_a z^2\]

\[= 3\log_a x + \frac{1}{2}\log_a y - 2\log_a z\]

Try it yourself: Simplify \(2\log 5 + \log 4\).

Click to reveal answer

\(\log 5^2 + \log 4 = \log 25 + \log 4 = \log(25 \times 4) = \log 100 = 2\).

Rick's Tip

The log laws work in both directions! You can expand \(\log(xy)\) into \(\log x + \log y\) (useful for solving equations), or condense \(\log x + \log y\) back into \(\log(xy)\) (useful for simplifying). Being fluent in both directions is a superpower for the IB exam.

Change of Base Rule

The change of base rule lets you convert a logarithm from one base to another:

\[\log_a x = \frac{\log_b x}{\log_b a}\]

The most common use is converting to base \(10\) or base \(e\) so you can use your calculator:

\[\log_a x = \frac{\log x}{\log a} = \frac{\ln x}{\ln a}\]

Worked Example 4: Calculate \(\log_3 20\) using common logarithms.

\[\log_3 20 = \frac{\log 20}{\log 3} = \frac{1.3010}{0.4771} \approx 2.727\]

This means \(3^{2.727} \approx 20\).

Try it yourself: Calculate \(\log_5 100\) using your calculator.

Click to reveal answer

\(\log_5 100 = \frac{\log 100}{\log 5} = \frac{2}{0.6990} \approx 2.861\).

Solving Logarithmic Equations

Solving log equations involves using the log-exponential conversion and logarithm laws to isolate the variable. There are two main types.

Type 1: Exponential equations solved with logarithms

These are the equations we couldn't solve in Chapter 7.

Worked Example 5: Solve \(2^x = 5\).

Take \(\log\) of both sides:

\[\log(2^x) = \log 5\]

\[x \log 2 = \log 5\]

\[x = \frac{\log 5}{\log 2} = \frac{0.6990}{0.3010} \approx 2.322\]

Worked Example 6: Solve \(e^{3x} = 40\).

Take \(\ln\) of both sides:

\[3x = \ln 40\]

\[x = \frac{\ln 40}{3} = \frac{3.6889}{3} \approx 1.230\]

Type 2: Equations with logarithmic terms

Worked Example 7: Solve \(\log_2(x + 3) = 5\).

Convert to exponential form:

\[x + 3 = 2^5 = 32\]

\[x = 29\]

Worked Example 8: Solve \(\log x + \log(x + 3) = 1\).

Use the product rule:

\[\log[x(x + 3)] = 1\]

\[x(x + 3) = 10^1 = 10\]

\[x^2 + 3x - 10 = 0\]

\[(x + 5)(x - 2) = 0\]

\[x = -5 \quad \text{or} \quad x = 2\]

Check: \(x = -5\) gives \(\log(-5)\), which is undefined. So \(x = 2\) is the only solution.

Rick's Common Mistake Alert

Always check your answers when solving log equations! Logarithms are only defined for positive arguments. If your solution makes any logarithm argument negative or zero, that solution is extraneous — reject it. This trap catches students every exam cycle.

Try it yourself: Solve \(3^{2x+1} = 54\).

Click to reveal answer

\(\log(3^{2x+1}) = \log 54\). \((2x+1)\log 3 = \log 54\). \(2x + 1 = \frac{\log 54}{\log 3} = \frac{1.7324}{0.4771} = 3.6309\). \(2x = 2.6309\). \(x \approx 1.315\).

Diagram: Logarithm Equation Solver

Logarithm Equation Solver

Type: microsim sim-id: logarithm-equation-solver
Library: p5.js
Status: Specified

Purpose: Visualize the solution of exponential and logarithmic equations by showing the graphs of both sides and highlighting the intersection as the solution.

Learning Objective: Students will solve exponential and logarithmic equations graphically and algebraically, verifying that solutions correspond to graph intersections (Apply L3 — solve, demonstrate).

Instructional Rationale: Graphical verification of algebraic solutions strengthens understanding of why logarithms "undo" exponentials. Seeing the intersection point reinforces that the solution satisfies both sides of the equation.

Visual elements:

• A coordinate grid showing two functions: the left-hand side and right-hand side of the equation
• The intersection point highlighted and its coordinates displayed
• The algebraic solution steps shown in a side panel
• Domain restrictions highlighted (e.g., the region where \(\log\) is undefined shown in gray)

Interactive controls:

• Dropdown: Select equation type from a bank: "\(2^x = 5\)," "\(e^{3x} = 40\)," "\(\log_2(x+3) = 5\)," "\(\log x + \log(x+3) = 1\)," "\(3^x = 10\)"
• Checkbox: "Show algebraic solution" — reveals step-by-step solution
• Checkbox: "Show domain restriction" — grays out region where logarithm is undefined
• Slider: Zoom level (to examine the intersection more closely)
• Button: "Next Equation"

Default: "\(2^x = 5\)" selected, algebraic solution hidden.

Canvas: Responsive, full width, 550px height

Logarithmic Scales

A logarithmic scale uses powers of \(10\) (or another base) instead of equal intervals. On a logarithmic scale, each step represents a multiplication rather than an addition.

Logarithmic scales are used when data spans many orders of magnitude — when the smallest and largest values differ by factors of thousands or millions.

Famous examples of logarithmic scales:

Scale	Measures	Each unit increase means...
Richter scale	Earthquake intensity	\(10\times\) more shaking
Decibel scale	Sound intensity	\(10\times\) more energy
pH scale	Acidity	\(10\times\) more H⁺ ions
Stellar magnitude	Star brightness	\(\approx 2.5\times\) dimmer

Worked Example 9: The Richter scale defines magnitude as \(M = \log\left(\frac{I}{I_0}\right)\) where \(I\) is intensity and \(I_0\) is a reference level. How many times more intense is a magnitude \(7\) earthquake compared to a magnitude \(5\)?

Difference in magnitude: \(7 - 5 = 2\).

Since each magnitude unit represents a factor of \(10\):

\[\frac{I_7}{I_5} = 10^2 = 100\]

A magnitude \(7\) earthquake is \(100\) times more intense than a magnitude \(5\).

Graphing Logarithms

Graphing logarithmic functions reveals a shape that is the mirror image of the corresponding exponential — reflected over the line \(y = x\).

Key features of \(f(x) = \log_a x\) (where \(a > 1\)):

• Passes through \((1, 0)\) (since \(\log_a 1 = 0\))
• Passes through \((a, 1)\) (since \(\log_a a = 1\))
• Vertical asymptote at \(x = 0\)
• Increases slowly — logarithmic growth is the opposite of exponential growth
• Domain: \((0, \infty)\); Range: \((-\infty, \infty)\)
• Always concave down

Try it yourself: Sketch the key features of \(f(x) = \log_2 x\). What is \(f(8)\)? What is \(f(\frac{1}{4})\)?

Click to reveal answer

\(f(8) = \log_2 8 = 3\). \(f(\frac{1}{4}) = \log_2 \frac{1}{4} = -2\). The graph passes through \((1, 0)\), \((2, 1)\), \((4, 2)\), \((8, 3)\) and approaches \(-\infty\) as \(x \to 0^+\).

Key Takeaways

This chapter introduced logarithmic functions — the inverse of exponentials:

• The log-exponential inverse relationship: \(a^y = x \iff y = \log_a x\)
• A logarithmic function \(f(x) = \log_a x\) has domain \((0, \infty)\) and vertical asymptote \(x = 0\)
• Common logarithm \(\log x = \log_{10} x\); natural logarithm \(\ln x = \log_e x\)
• Three logarithm laws: product (\(\log mn = \log m + \log n\)), quotient (\(\log \frac{m}{n} = \log m - \log n\)), power (\(\log m^k = k\log m\))
• The change of base rule: \(\log_a x = \frac{\log x}{\log a}\) lets you use your calculator for any base
• Solving log equations: use log-exponential conversion and always check for extraneous solutions
• Logarithmic scales (Richter, decibel, pH) compress vast ranges into manageable numbers
• Graphing logarithms produces a slowly-increasing curve that mirrors the exponential

Nice Work!

You've now mastered both sides of the exponential-logarithm mirror! Exponentials and logarithms are one of the great partnerships in mathematics — like multiplication and division, they complete each other. With these tools, you can solve any exponential equation and understand phenomena that span from earthquakes to sound waves. Trust the process — and the math!