Logarithmic functions for MATH 140

The Spotify Problem

Imagine you're an engineer at a music streaming platform. Your boss wants a simple 5-tier ranking system for songs: Bronze, Silver, Gold, Platinum, and Diamond.

The data team reports that song play counts range from 1 play (your roommate's SoundCloud demo) to 10 billion plays (global mega-hits like "Blinding Lights").

"Easy," you think. "We'll divide it evenly — 2 billion plays per tier."

Then you test it:

• Your roommate's demo (12 plays) → Bronze
• A local band's regional hit (50,000 plays) → Bronze
• A chart-topping single (5 million plays) → Bronze
• A viral TikTok song (500 million plays) → Bronze

Wait... 99.9% of all songs are Bronze? The tier system is useless.

The problem: these play counts span 10 orders of magnitude (from 1 to 10,000,000,000). A linear scale crushes all the "small" values together.

The Fix: Think Multiplicatively

What if each tier represented a ×100 jump instead of a +2 billion jump?

Now:

• Your roommate's demo (12 plays) → Bronze
• Local band's regional hit (50,000 plays) → Gold
• Chart-topping single (5 million plays) → Platinum
• Viral TikTok song (500 million plays) → Diamond

Each tier represents a meaningful level of success. Going from 100 to 10,000 plays is the same "jump" as going from 1 million to 100 million — both are ×100.

This is logarithmic thinking. Instead of asking "how much more?", we ask "how many times more?"

The Connection to Exponentials

In the last lesson, you saw exponential growth: a penny doubled daily becomes $10.7 million. The question was "start with a base, apply the exponent, what do you get?"

But what if you flip the question?

Exponential: "A song has 10,000 plays. If plays increase 10× per tier, what tier is it?"
Logarithmic: "I know the result (10,000) and the multiplier (10). What's the tier number?"

That tier number — the exponent you need — is exactly what a logarithm tells you.

$$\log_{10}(10{,}000) = 4$$

The song is in tier 4. Logarithms answer: "What exponent do I need to reach this number?"

What is a Logarithm?

A logarithm answers the question: "What exponent do I need?"

$$\log_a(x) = y \quad \text{means} \quad a^y = x$$

Read $\log_a(x)$ as "log base $a$ of $x$."

The key insight: Logarithms and exponentials are two ways of expressing the same relationship:

Think of it this way:

• Exponential: You know the exponent, you want the result
• Logarithm: You know the result, you want the exponent

The Two Most Important Logarithms

While you can have a logarithm with any positive base (except 1), two bases dominate:

Common Logarithm (Base 10)

$$\log_{10}(x) = \log(x)$$

When you see "$\log$" with no base written, it usually means base 10. This is the "common logarithm."

Why base 10? Because we use a decimal number system!

• $\log(10) = 1$ (one zero)
• $\log(100) = 2$ (two zeros)
• $\log(1000) = 3$ (three zeros)
• $\log(1{,}000{,}000) = 6$ (six zeros)

The common logarithm essentially counts "how many digits minus one" or "what order of magnitude."

Natural Logarithm (Base $e$)

$$\log_e(x) = \ln(x)$$

The natural logarithm uses base $e \approx 2.71828$ and is written as "$\ln$" (from the Latin logarithmus naturalis).

Why is this "natural"? In calculus, $\ln(x)$ has the beautiful property:

$$\frac{d}{dx}[\ln(x)] = \frac{1}{x}$$

No other logarithm has such a clean derivative. This makes $\ln(x)$ the go-to logarithm for calculus, just like $e^x$ is the go-to exponential.

Converting Between Forms

The ability to switch between exponential and logarithmic forms is essential. Practice this until it's automatic:

From Exponential to Logarithmic

$$a^y = x \quad \Longrightarrow \quad \log_a(x) = y$$

The base stays the base. The exponent becomes the answer. The result becomes the input.

Examples:

• $3^4 = 81 \quad \Longrightarrow \quad \log_3(81) = 4$
• $e^2 \approx 7.389 \quad \Longrightarrow \quad \ln(7.389) \approx 2$
• $10^{-1} = 0.1 \quad \Longrightarrow \quad \log(0.1) = -1$

From Logarithmic to Exponential

$$\log_a(x) = y \quad \Longrightarrow \quad a^y = x$$

Examples:

• $\log_2(32) = 5 \quad \Longrightarrow \quad 2^5 = 32$
• $\ln(1) = 0 \quad \Longrightarrow \quad e^0 = 1$
• $\log_5(125) = 3 \quad \Longrightarrow \quad 5^3 = 125$

The Graph of a Logarithmic Function

Let's visualize $f(x) = \log_2(x)$ by building a table:

Notice the shape: it rises steeply at first, then grows more and more slowly. This is the opposite of exponential growth!

Logarithms and Exponentials are Inverses

Here's the profound relationship: logarithms and exponentials undo each other.

$$\log_a(a^x) = x \quad \text{and} \quad a^{\log_a(x)} = x$$

Think of it like this:

• $\sqrt{\ }$ undoes squaring: $\sqrt{x^2} = x$
• $\log_a$ undoes $a^x$: $\log_a(a^x) = x$

Graphically, this means the graphs of $y = a^x$ and $y = \log_a(x)$ are reflections of each other across the line $y = x$.

This reflection swaps:

• Domain and range
• Horizontal and vertical asymptotes
• The roles of $x$ and $y$

Key Properties of Logarithmic Functions

For $f(x) = \log_a(x)$ where $a > 0$ and $a \neq 1$:

Why can't we take the log of zero or negative numbers?

Think about it: $\log_2(0) = ?$ would mean $2^? = 0$. But no power of 2 ever equals zero!

And $\log_2(-8) = ?$ would mean $2^? = -8$. But $2^x$ is always positive!

Notice that ALL these curves pass through $(1, 0)$. That's because $a^0 = 1$ for any positive $a$, so $\log_a(1) = 0$.

Special Values to Memorize

These show up constantly — know them cold:

Logarithmic vs. Exponential Growth

Logarithms grow incredibly slowly compared to other functions.

To get $\log_2(x) = 20$, you need $x = 2^{20} = 1{,}048{,}576$ — over a million!

This slow growth is actually useful! It's why we use logarithmic scales:

• Decibels (sound intensity) — lets us compare a whisper to a jet engine
• Richter scale (earthquakes) — a magnitude 8 is 10× more intense than magnitude 7
• pH scale (acidity) — each unit is a 10× change in hydrogen ion concentration

Why Logarithms Matter in Real Life

The Decibel Scale

Sound intensity varies enormously. A jet engine is about 1,000,000,000,000 times more intense than the quietest sound you can hear. That's unwieldy!

Instead, we use decibels: $dB = 10 \log\left(\frac{I}{I_0}\right)$

Logarithms compress huge ranges into manageable numbers!

The Richter Scale

Earthquake magnitude: $M = \log\left(\frac{A}{A_0}\right)$

Each whole number increase means 10× more ground shaking. A magnitude 8 earthquake releases about 32 times more energy than a magnitude 7.

Computer Science

Logarithms appear everywhere in algorithms:

• Binary search takes $\log_2(n)$ steps to find an item in a sorted list of $n$ items
• Efficient sorting algorithms take about $n \log(n)$ operations

If you have 1 billion items, binary search finds any item in about $\log_2(1{,}000{,}000{,}000) \approx 30$ steps. That's the power of logarithmic efficiency!

Practice: Evaluate and Convert

Example 6

Problem: What is the domain of $f(x) = \log_2(x - 3)$?

Show Solution

The input to a logarithm must be positive:

$x - 3 > 0$

$x > 3$

$$\boxed{\text{Domain: } (3, \infty)}$$

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Common Mistakes and Misunderstandings

❌ Mistake: Taking log of zero or negative numbers

Wrong: "$\log(-4) = ?$" or "$\ln(0) = ?$"

Why it's wrong: Logarithms are only defined for positive inputs. There's no real number $y$ such that $10^y = -4$ or $e^y = 0$.

Correct: The domain of any logarithm is $(0, \infty)$. If someone asks for $\log(-4)$, the answer is "undefined" (in the real numbers).

❌ Mistake: Forgetting that $\log_a(1) = 0$, not 1

Wrong: "$\log_5(1) = 1$"

Why it's wrong: $\log_5(1)$ asks "5 to what power equals 1?" Since $5^0 = 1$, the answer is 0.

Correct: $\log_a(1) = 0$ for any valid base $a$. Remember: any number to the zero power is 1.

❌ Mistake: Confusing $\log_a(x)$ with $\log(a) \cdot x$

Wrong: Treating $\log_2(8)$ as $\log(2) \times 8$

Why it's wrong: The subscript indicates the base, not multiplication. $\log_2(8)$ means "log base 2 of 8."

Correct: $\log_2(8) = 3$ because $2^3 = 8$. It has nothing to do with $\log(2) \times 8$.

❌ Mistake: Thinking $\log(a + b) = \log(a) + \log(b)$

Wrong: "$\log(3 + 5) = \log(3) + \log(5)$"

Why it's wrong: Logarithm of a sum does NOT split up. This is a common trap!

Correct: $\log(3 + 5) = \log(8) \approx 0.903$, but $\log(3) + \log(5) = \log(3 \times 5) = \log(15) \approx 1.176$. These are different!

The actual rule is: $\log(a \cdot b) = \log(a) + \log(b)$ (logarithm of a product splits into a sum).

❌ Mistake: Wrong direction when converting forms

Wrong: Converting $\log_3(81) = 4$ to "$3^{81} = 4$"

Why it's wrong: The number inside the log becomes the result of the exponential, not the exponent.

Correct: $\log_3(81) = 4$ converts to $3^4 = 81$. The base stays the base, the answer (4) becomes the exponent, and the input (81) becomes the result.

Summary: Exponentials vs. Logarithms

They are inverse functions — each undoes the other:

• $\log_a(a^x) = x$
• $a^{\log_a(x)} = x$