📝 Blogarithmic Functions

Where MAT 172 meets the real world — pH, log reductions, decibels, and more

The Lab MAT 172 BIO 111
You've been doing logarithm math at work for years — every pH reading, every log reduction calculation, every serial dilution. You just didn't know it had a name. Now you do.
The big idea: A logarithm asks the inverse question of an exponent. If 10² = 100, then log₁₀(100) = 2. You're asking: "What power do I raise the base to in order to get this number?"
If  10² = 100  then  log₁₀(100) = 2
Two ways of saying the exact same thing
2.0
Exponential form
10² = 100
Logarithmic form
log₁₀(100) = 2
pH 2 is battery acid — 100× more acidic than pH 4 (lemon juice). Each pH unit is a 10× difference in H⁺ concentration.
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Exponents vs. logarithms — the relationship
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Exponential and logarithmic functions are inverses of each other — they undo each other the same way multiplication and division do. The general relationship is:

b^x = y  ⟺  log_b(y) = x
The base b must be positive and not equal to 1. The argument (y) must be positive — you can't take the log of zero or a negative number. This is why log scales only work for positive quantities like concentration, intensity, or population size.
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Why logarithmic scales exist
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Some quantities in nature span such enormous ranges that a linear scale becomes useless. Sound intensity ranges from a whisper (0.000001 W/m²) to a jet engine (100 W/m²) — a range of 100 million. If you tried to plot this on a linear graph, the whisper would be invisible. A log scale compresses these ranges into something human-readable. Any time you see "per unit" or "per decade" thinking — that's logarithmic. The human ear, eye, and even our sense of earthquake magnitude all perceive things logarithmically.
💡 This is also why a pH change from 6.8 (milk) to 4.5 (yogurt) sounds small but represents a 200× increase in H⁺ concentration. The numbers compress the reality.
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pH — the logarithm you measure every day
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pH = −log₁₀[H⁺] where [H⁺] is the hydrogen ion concentration in mol/L. The negative sign flips the scale so lower H⁺ concentration = higher pH (more basic). Every one-unit change in pH = a 10× change in H⁺ concentration.
Milk pH ≈ 6.7
[H⁺] = 10⁻⁶·⁷ ≈ 2×10⁻⁷ mol/L
Yogurt pH ≈ 4.5
[H⁺] = 10⁻⁴·⁵ ≈ 3×10⁻⁵ mol/L
That 2.2 pH unit drop from milk to yogurt = a ~160× increase in H⁺ concentration. That's what lactic acid bacteria produce — not a small shift. A logarithmic one.
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Log reductions — food safety math
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A log reduction describes how much you've reduced a microbial population using powers of 10. A 1-log reduction = 90% kill, 2-log = 99%, 3-log = 99.9%, and so on. Pasteurization of milk is typically a 5-log reduction of the target pathogen — meaning if you started with 100,000 CFU/mL of the pathogen, you'd end with ≤1 CFU/mL.
Log reduction = log₁₀(N₀ / N)
N₀ = initial count · N = final count
Example: Initial count 10⁶ CFU/g, final count 10¹ CFU/g → log₁₀(10⁶/10¹) = log₁₀(10⁵) = 5-log reduction. This is the actual math behind your validation protocols.
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Serial dilutions — log in the lab
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A serial dilution is a series of stepwise dilutions, each typically 1:10 (one part sample in nine parts diluent). Each dilution step is one log unit. If you dilute 10⁻¹, 10⁻², 10⁻³ and plate the 10⁻³ dilution and count 45 colonies, the original concentration was 45 × 10³ = 45,000 CFU/mL.
Original
10⁰ = 1
undiluted
10⁻¹
÷10
1:10
10⁻²
÷100
1:100
10⁻³
÷1000
1:1000
Each step is one log unit. Petrifilm plating at 10⁻³ and multiplying by 1000 to get back to original concentration — that multiplication is undoing the logarithmic dilution.
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Decibels — sound on a log scale
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Sound intensity is measured in decibels (dB): dB = 10 × log₁₀(I / I₀) where I₀ is the threshold of human hearing (10⁻¹² W/m²). A 10 dB increase = 10× more intense sound. A 20 dB increase = 100× more intense. But here's the counterintuitive part — our perception of loudness doesn't scale linearly with intensity. We perceive a 10× more intense sound as roughly twice as loud. Logs model our perception better than linear scales.
30 dB
Whisper
85 dB
Heavy traffic
120 dB
Pain threshold
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Richter scale — earthquakes and logs
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The Richter scale is logarithmic: each whole number increase represents a 10× increase in ground motion amplitude and roughly 31.6× more energy released. A magnitude 6.0 earthquake releases ~1,000× more energy than a 4.0 (because 31.6² ≈ 1000). This is why a 9.0 earthquake (like the 2011 Tōhoku event) is incomprehensibly more powerful than a 6.0 — not 1.5× more, but thousands of times more. The Richter scale makes these enormous differences expressible as single-digit numbers.
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Microbial growth — log phase in culture
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Bacterial growth during the exponential (log) phase is — as the name says — logarithmic. Cell counts double every generation time. If you plot cell count on a linear scale you get a curve that rockets upward. Plot it on a log scale (log CFU/mL vs time) and it becomes a straight line — much easier to read, interpret, and compare growth rates between conditions.
💡 S. thermophilus doubling time at 42°C ≈ 20–30 minutes. After 3 hours that's ~6–9 doublings = 64–512× cell increase. In log terms: 1.8–2.7 log increase. This is why your incubation window matters — you're riding the log phase curve.
Four rules, infinite applications. Learn these cold — they show up in every log problem on your exam.
Product rule
log_b(xy) = log_b(x) + log_b(y)
log(100 × 10) = log(100) + log(10) = 2 + 1 = 3 ✓ (because 100×10=1000=10³)
Quotient rule
log_b(x/y) = log_b(x) − log_b(y)
log(1000/10) = log(1000) − log(10) = 3 − 1 = 2 ✓ (because 1000/10=100=10²)
Power rule
log_b(xⁿ) = n · log_b(x)
log(10³) = 3 · log(10) = 3 · 1 = 3 ✓ This is how log reductions work: log(10⁵) = 5.
Change of base
log_b(x) = log(x) / log(b)
log₂(8) = log(8)/log(2) = 0.903/0.301 = 3 ✓ (because 2³=8). Use this on your calculator.
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Special values to memorize
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log_b(1) = 0
because b⁰ = 1 for any base
log_b(b) = 1
because b¹ = b
log_b(bⁿ) = n
log and exponent cancel out
b^(log_b(x)) = x
exponent and log cancel out
The last two are the inverse relationship in action — logs and exponentials undo each other. If you ever see a log inside an exponent or vice versa, your first instinct should be to cancel them.
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Common mistakes — don't do these
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✗  log(x + y) ≠ log(x) + log(y)
The product rule is for log(x·y), not log(x+y). There is no rule for sum of arguments.
✗  log(x/y) ≠ log(x) / log(y)
Quotient rule gives log(x) − log(y), not division of logs. Change of base is the only time you divide logs.
✗  log(xⁿ) ≠ log(x)ⁿ
Power rule brings the exponent OUT FRONT as a coefficient: n·log(x). It does not raise the whole log to the nth power.
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Solving log equations — the method
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Two main strategies depending on the equation form:

Strategy 1 — Convert to exponential form: If log_b(x) = c, rewrite as b^c = x. Isolate the log, then exponentiate.

Strategy 2 — Get same base, equate arguments: If log_b(A) = log_b(B), then A = B.

Example: Solve log₁₀(x) = 3.5
→ 10^3.5 = x
→ x = 10^3.5 ≈ 3162

Example: Solve log₁₀(x) + log₁₀(4) = 2
→ log₁₀(4x) = 2  (product rule)
→ 4x = 10² = 100
→ x = 25
Key shape: Logarithmic graphs pass through (1, 0) always, approach x = 0 as a vertical asymptote, and grow without bound but slower and slower. Drag the sliders to see how the base and transformations change the curve.
10
1
0
0
y = log₁₀(x)
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Key features of log graphs
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Domain: x > 0 (x must be positive — you can't log a negative or zero)
Range: all real numbers (−∞, ∞)
x-intercept: always (1, 0) since log_b(1) = 0
Vertical asymptote: x = 0 (the y-axis). The graph approaches but never crosses.
b > 1: increasing function (most common — log₁₀, ln)
0 < b < 1: decreasing function (rare in practice)
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Log vs exponential — mirror images
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The graph of y = log_b(x) is the graph of y = b^x reflected across the line y = x. They are perfect inverses — every point (a, b) on the exponential graph has a corresponding point (b, a) on the log graph. This is why they have swapped domain/range: exponentials have domain = all reals, range = positive reals; logs have domain = positive reals, range = all reals. On your exam you may be asked to graph one given the other — just reflect across y = x.
💡 Try setting a = −1 in the graph above. The curve flips vertically — this is a reflection across the x-axis. Combine transformations to build any log function your exam might throw at you.

Quick reference

b^x = y ⟺ log_b(y) = x
log(xy) = log x + log y
log(x/y) = log x − log y
log(xⁿ) = n · log x
log_b(x) = log x / log b
log without a base = log₁₀ (common log). ln = log base e ≈ 2.718 (natural log). Your calculator has both buttons.

Key vocabulary

Logarithm
The exponent to which a base must be raised to produce a given number. log_b(y) = x means b^x = y.
Common log
log base 10, written as log(x). Used for pH, decibels, Richter scale, and log reductions.
Natural log (ln)
log base e (≈ 2.718). Used in calculus, population growth models, and half-life calculations.
Log reduction
log₁₀(N₀/N) — how many powers of 10 you reduced a microbial population. 5-log = 99.999% kill.
Vertical asymptote
A vertical line the graph approaches but never touches. For y = log(x), it's x = 0 (the y-axis).
Inverse functions
Functions that undo each other. log_b(x) and b^x are inverses — their graphs reflect across y = x.

Exam reminders

Domain of log functions is x > 0 only. Always check this when solving equations — extraneous solutions sneak in.
log(1) = 0 always. log(b) = 1 always. These show up constantly.
There is no product rule for addition inside a log. log(x + y) cannot be simplified. Full stop.
To solve a log equation: isolate the log, then exponentiate both sides with the base to cancel the log.
A 5-log reduction means 99.999% kill. USDA and FDA use this language in pathogen reduction standards — and now so do you.

Where logs live in your world

pH meter: −log₁₀[H⁺]
Petrifilm math: log₁₀(N₀/N)
Serial dilution: 10⁻¹, 10⁻², 10⁻³
Log growth phase: CFU vs time on log scale