Applications of Logarithms: Explained with Examples

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A logarithm answers one simple question: to what power must a base number be raised to produce a given value? Saying that the logarithm of 32 to base 2 is 5 is exactly the same as saying that 2 raised to the power 5 equals 32. In symbols, log_b(x) = y means precisely b^y = x, where the base b is a positive number different from 1, and x is positive. The logarithm undoes exponentiation, just as subtraction undoes addition.

Logarithms matter because they shrink gigantic ranges into numbers humans can handle. The energy released by earthquakes, the intensity of sounds, and the concentration of acid in a solution can differ by factors of millions or billions. A logarithm converts multiplication into addition: instead of multiplying 1,000 by 10,000, a scientist adds 3 and 4. That is why logarithmic scales appear across science and finance.

Key Concepts

Definition: log_b(x) = y exactly when b^y = x, with b > 0, b not equal to 1, and x > 0.
Common logarithm: base 10, written log x. Natural logarithm: base e (about 2.71828), written ln x.
Product law: log_b(xy) = log_b x + log_b y. The log of a product is the sum of the logs.
Quotient law: log_b(x/y) = log_b x - log_b y.
Power law: log_b(x^k) = k times log_b x. Exponents slide down in front of the log.
Change of base: log_b x = log_c x / log_c b.

Formulas

pH: pH = -log_10[H+], where [H+] is the hydrogen-ion concentration in moles per litre.
Richter scale: M = log_10(A/A_0), comparing a quake's wave amplitude A with a reference amplitude A_0. Each whole-number step means ten times the amplitude.
Decibels: sound level = 10 times log_10(I/I_0), with I_0 the hearing threshold.
Doubling time: t = log 2 / log(1 + r) gives the number of periods needed for an investment growing at rate r per period to double.

Worked Example

Money is invested at 5% compound interest per year. After how many years does it double? Doubling means (1.05)^t = 2. Take the common logarithm of both sides: log((1.05)^t) = log 2. By the power law, t times log 1.05 = log 2, so t = log 2 / log 1.05. Now log 2 is about 0.3010 and log 1.05 is about 0.02119, giving t = 0.3010 / 0.02119 = 14.2 years. Checking: 1.05^14 is about 1.98 and 1.05^15 is about 2.08, so the money doubles between years 14 and 15, matching 14.2.

Common Misconceptions

log of a negative number: many students assume log(-5) is some negative value. In real-number mathematics the logarithm is defined only for positive arguments, so log(-5) has no real value at all.
log(a + b) = log a + log b: false. The addition law works for products, log(ab) = log a + log b, never for sums.
Reversed meaning: log_b x = y does NOT mean x^y = b. It means b^y = x. The base is raised to the logarithm to recover x.

Real-World Applications

Chemistry: the pH scale rates acidity from 0 to 14 using base-10 logarithms of ion concentration.
Seismology: the Richter scale compares earthquake energies logarithmically.
Acoustics: loudness is measured in decibels, from a whisper at about 30 dB to a jet engine near 140 dB.
Finance: compound interest, doubling time, and continuous growth are all modelled with logarithms.

Logarithms turn repeated multiplication into simple addition, taming quantities that span many orders of magnitude. Master the three log laws, remember that only positive numbers have real logarithms, and every logarithmic scale becomes readable.