Decimal logarithms – Arithmetic (Senior 4 Basic Mathematics) | Ganzaa

Arithmetic

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Senior 4 · Basic Mathematics

Arithmetic

5 lessons

Lesson 4 of 5

Decimal logarithms

25 min

Counting the zeros

Rwanda's census counted 4 831 527 people in 1978 and 13 246 394 in 2022. Both numbers have 7 digits: both lie between $10^{6}$ and $10^{7}$ . The exponent of 10 tells you the size of a number at a glance, and the decimal logarithm makes that exponent exact for every positive number: $\log 13\,246\,394 \approx 7.12$ , so $13\,246\,394 \approx 10^{7.12}$ . Logarithms turn multiplication into addition and powers into multiplication, which is why, for more than three centuries before electronic calculators, scientists and engineers used printed tables of logarithms and slide rules. Today you need them for one main job: solving equations where the unknown is in an exponent, such as "how many years until my savings double?".

Definition

Decimal (common) logarithm: for a positive number $x$ , $\log x$ is the exponent to which 10 must be raised to give $x$ :

\log x = y \iff 10^y = x \qquad (x \gt 0).

It is also written $\log_{10} x$ . The logarithm of 0 or of a negative number is not defined, because $10^{y}$ is always positive.

Examples: $\log 1000 = 3$ since $10^{3} = 1000$ ; $\log 0.01 = -2$ since $10^{-2} = 0.01$ ; $\log 1 = 0$ ; $\log 10 = 1$ ; $\log \sqrt{10} = \frac{1}{2}$ .

The graph of $y = \log x$

The graph shows the main facts: $\log x$ is negative for $0 \lt x \lt 1$ , zero at $x = 1$ , positive for $x \gt 1$ , and always increasing, so a bigger number has a bigger logarithm. It grows very slowly: going from 1 to 10 adds only 1 to the logarithm, and so does going from 10 to 100.

Important

Laws of logarithms. For $a, b \gt 0$ and any real $n$ :

\log(ab) = \log a + \log b, \qquad \log\frac{a}{b} = \log a - \log b, \qquad \log a^n = n \log a,

\log 10^n = n, \qquad 10^{\log a} = a .

Why the first law holds: write $a = 10^{p}$ and $b = 10^{q}$ , so $p = \log a$ and $q = \log b$ . Then $ab = 10^p \times 10^q = 10^{p+q}$ , so $\log(ab) = p + q = \log a + \log b$ . The other laws come the same way from the laws of powers.

Worked example 1: using the laws without a calculator

Evaluate (a) $\log 4 + \log 25$ (b) $\log 500 - \log 5$ (c) $3\log 2 + \log 125$ . (d) Write $\log\frac{x^2\sqrt{y}}{z}$ in terms of $\log x$ , $\log y$ and $\log z$ .

Solution

$\log(4 \times 25) = \log 100 = 2$ .
$\log\frac{500}{5} = \log 100 = 2$ .
$\log 2^3 + \log 125 = \log(8 \times 125) = \log 1000 = 3$ .
$\log x^2 + \log y^{\frac{1}{2}} - \log z = 2\log x + \frac{1}{2}\log y - \log z$ .

Worked example 2: an unknown in the exponent

Solve (a) $10^{x} = 50$ (b) $3^{x} = 20$ , to 3 decimal places.

Solution

By the definition, $x = \log 50 \approx 1.699$ .
Take the logarithm of both sides: $\log 3^x = \log 20$ , so $x\log 3 = \log 20$ and

x = \frac{\log 20}{\log 3} = \frac{1.30103}{0.47712} \approx 2.727 .

Check: $3^{2.727} \approx 20.0$ . ✓

Worked example 3: logarithmic equations

Solve (a) $\log(x + 3) + \log x = 1$ (b) $2\log x = \log(5x - 6)$ .

Solution

Domain: $x + 3 \gt 0$ and $x \gt 0$ , so $x \gt 0$ . Combine: $\log[x(x + 3)] = 1$ , so $x (x + 3) = 10^{1} = 10$ , $x^{2} + 3 x - 10 = 0$ , $(x + 5) (x - 2) = 0$ . The root $x = - 5$ is outside the domain, so the only solution is $x = 2$ . Check: $\log 5 + \log 2 = \log 10 = 1$ . ✓
Domain: $x \gt 0$ and $5x - 6 \gt 0$ , so $x \gt \frac{6}{5}$ . Then $\log x^2 = \log(5x - 6)$ , so $x^{2} = 5 x - 6$ , $(x - 2) (x - 3) = 0$ . Both $x = 2$ and $x = 3$ are greater than $\frac{6}{5}$ , so both are solutions.

Worked example 4: how long to double?

NISR reports that Rwanda's population grew at an average of 2.3% a year between 2012 and 2022. If that rate continued, how many years would a population take to double?

Solution

Each year multiplies the population by 1.023, so we need ${1.023}^{n} = 2$ . Taking logarithms, $n\log 1.023 = \log 2$ :

n = \frac{\log 2}{\log 1.023} = \frac{0.30103}{0.0098756} \approx 30.5 \text{ years}.

This is a "what if" calculation, not a forecast: growth rates change over time.

Warning

The laws are about products, not sums. $\log(a + b)$ is not $\log a + \log b$ , and $\frac{\log a}{\log b}$ is not $\log\frac{a}{b}$ ( $\frac{\log 100}{\log 10} = 2$ , but $\log 10 = 1$ ). Always state the domain of a logarithmic equation first, and reject any root that makes the inside of a logarithm zero or negative.

Tip

Memorise $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ . With the laws you then get many others: $\log 5 = \log\frac{10}{2} = 1 - 0.301 = 0.699$ , $\log 6 = 0.301 + 0.477 = 0.778$ , $\log 8 = 3 \times 0.301 = 0.903$ . The number of digits of a whole number $N$ is the whole-number part of $\log N$ , plus 1.

JensenMath: Logarithms, Top 10 Must Knows. Watch the link between logarithms and exponents, the three laws, and solving equations with the unknown in the exponent.

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Practice

Try each one before opening the answers. They get harder as you go.

Write in logarithm form: (a) $10^4 = 10\,000$ (b) $10^{-3} = 0.001$ . Write in power form: (c) $\log 316 \approx 2.5$ .
Evaluate without a calculator: (a) $\log 0.0001$ (b) $\log 20 + \log 5$ (c) $\log 8 - \log 80$ .
Given $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ , find (a) $\log 12$ (b) $\log 1.5$ (c) $\log 0.9$ .
Solve (a) $10^{2x} = 300$ (b) $2^{x} = 30$ , to 3 significant figures.
Solve $\log(2x + 4) = 2$ .
Solve $\log x + \log(x - 15) = 2$ .
Savings grow by 8% a year. After how many whole years will they first be more than 1.5 times the starting amount?

Show answersHide answers

(a) $\log 10\,000 = 4$ (b) $\log 0.001 = -3$ (c) $10^{2.5} \approx 316$ .
(a) $- 4$ (b) $\log 100 = 2$ (c) $\log\frac{8}{80} = \log 0.1 = -1$ .
(a) $\log(4 \times 3) = 2(0.301) + 0.477 = 1.079$ (b) $\log\frac{3}{2} = 0.477 - 0.301 = 0.176$ (c) $\log\frac{9}{10} = 2(0.477) - 1 = -0.046$ .
(a) $2x = \log 300 \approx 2.477$ , $x \approx 1.24$ (b) $x = \frac{\log 30}{\log 2} \approx 4.91$ .
$2 x + 4 = 100$ , $x = 48$ .
Domain $x \gt 15$ . $x (x - 15) = 100$ , $x^{2} - 15 x - 100 = 0$ , $(x - 20) (x + 5) = 0$ ; $x = 20$ (reject $- 5$ ).
$1.08^n \gt 1.5$ : $n \gt \frac{\log 1.5}{\log 1.08} \approx 5.27$ , so after 6 years.

Quick check

Key points

$\log x = y \iff 10^y = x$ , defined only for $x \gt 0$ ; $\log 1 = 0$ , $\log 10 = 1$ .
$\log(ab) = \log a + \log b$ , $\log\frac{a}{b} = \log a - \log b$ , $\log a^n = n\log a$ .
To solve $a^{x} = b$ , take logarithms: $x = \frac{\log b}{\log a}$ .
In log equations, state the domain first and reject roots outside it.

Sources and credits

Written by Ganzaa: the explanations, the graph, the worked examples, the practice items and the questions. Census totals (1978, 2022) and the 2012–2022 average annual growth rate are from Table 4 of NISR's Fifth Population and Housing Census, Rwanda 2022: Main Indicators, CC BY 4.0.
Photo: Rechenschieber (slide rule), Berthold Werner, Wikimedia Commons, CC BY-SA 4.0.
Video: LOGARITHMS Top 10 Must Knows (ultimate study guide), JensenMath, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Counting the zeros

2The graph of y=log⁡x y = \log x y=logx

Counting the zeros

The graph of $y = \log x$