Number & Algebra

Number Bases, Surds and Logarithms

SSS 1 · SSS 2 · SSS 3

✓By the end of this lesson students will be able to convert numbers from one base to another.
✓By the end of this lesson students will be able to perform basic arithmetic operations on numbers in different bases.
✓By the end of this lesson students will be able to simplify and rationalise expressions involving surds.
✓By the end of this lesson students will be able to apply the laws of logarithms to simplify and solve logarithmic equations.
✓By the end of this lesson students will be able to use common logarithms and antilogarithms for calculations.

Key concepts

Number Bases

A number base is the number of unique digits, including zero, that a number system uses to represent numbers. The most common base is base 10 (decimal system), which uses digits 0-9. Other common bases include base 2 (binary), base 8 (octal), and base 16 (hexadecimal).

Conversion from Other Bases to Base 10

To convert a number from any base 'b' to base 10, multiply each digit by the base raised to the power of its position (starting from 0 for the rightmost digit).

d_n d_{n-1} ... d_1 d_0_b = d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0_{10}

Conversion from Base 10 to Other Bases

To convert a number from base 10 to another base 'b', repeatedly divide the base 10 number by 'b' and record the remainders. The new number is formed by writing the remainders from bottom to top.

Arithmetic Operations in Other Bases

Addition, subtraction, multiplication, and division can be performed in other bases using similar methods to base 10, but 'carrying over' or 'borrowing' occurs when the sum or difference reaches or exceeds the base value.

Surds

A surd is an irrational number that can be expressed as the root of an integer or a fraction, but whose exact value cannot be expressed as a simple fraction. Examples include √2, √3, √5. If the root can be simplified to a rational number (e.g., √4 = 2), it is not a surd.

Simplification of Surds

To simplify a surd, find the largest perfect square factor of the number under the square root sign. Then, take the square root of that factor and leave the remaining factor under the square root.

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

Rationalisation of the Denominator

Rationalising the denominator means removing any surds from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by an appropriate surd or its conjugate.

For \frac{a}{\sqrt{b}}, multiply by \frac{\sqrt{b}}{\sqrt{b}}. For \frac{a}{b + \sqrt{c}}, multiply by \frac{b - \sqrt{c}}{b - \sqrt{c}}.

Logarithms

A logarithm is the power to which a base must be raised to produce a given number. It is the inverse operation of exponentiation. If b^x = N, then log_b N = x.

If b^x = N, then log_b N = x

Laws of Logarithms

These are fundamental rules for manipulating logarithmic expressions.

1. log_b (MN) = log_b M + log_b N\n2. log_b (M/N) = log_b M - log_b N\n3. log_b (M^p) = p log_b M\n4. log_b b = 1\n5. log_b 1 = 0\n6. log_b M = \frac{log_a M}{log_a b} (Change of Base Law)

Common Logarithms and Antilogarithms

Common logarithms are logarithms with base 10 (written as log N). Antilogarithm (antilog) is the inverse of logarithm; if log N = x, then N = antilog x = 10^x. These are used for calculations involving multiplication, division, powers, and roots.

log N = log_{10} N; antilog x = 10^x

Key facts to remember

1In base 'b', the digits used are 0, 1, 2, ..., (b-1).
2To convert from base 'b' to base 10, use the place value expansion method.
3To convert from base 10 to base 'b', use the repeated division method.
4A surd is an irrational root, e.g., √2, √3. √4 is not a surd because it simplifies to 2.
5Only like surds can be added or subtracted.
6To rationalise a denominator with a single surd (e.g., √a), multiply by √a/√a.
7To rationalise a denominator with a binomial surd (e.g., a + √b), multiply by its conjugate (a - √b).
8The relationship between indices and logarithms is: b^x = N <=> log_b N = x.
9Key laws of logarithms: log(MN) = log M + log N, log(M/N) = log M - log N, log(M^p) = p log M.
10log_b b = 1 and log_b 1 = 0 for any valid base b.

Worked examples

Example 1

1. Convert 432_5 to base 10.

IIdentify the place values for each digit: 4 is in the 5^2 position, 3 in the 5^1 position, and 2 in the 5^0 position.

IIMultiply each digit by its corresponding place value:

III4 \times 5^2 + 3 \times 5^1 + 2 \times 5^0

IV= 4 \times 25 + 3 \times 5 + 2 \times 1

V= 100 + 15 + 2

VI= 117

Answer

432_5 = 117_10

Always remember that any number raised to the power of 0 is 1.

Example 2

2. Convert 157_10 to base 8.

IDivide 157 by 8:

II157 \div 8 = 19 remainder 5

IIIDivide 19 by 8:

IV19 \div 8 = 2 remainder 3

VDivide 2 by 8:

VI2 \div 8 = 0 remainder 2

VIIRead the remainders from bottom to top: 235.

Answer

157_10 = 235_8

The division continues until the quotient is 0.

Example 3

3. Evaluate 1011_2 + 110_2.

IAlign the numbers by their place values:

II 1011_2

III+ 110_2

IV--------

VAdd the rightmost digits: 1 + 0 = 1.

VIAdd the next digits: 1 + 1 = 2. In base 2, 2 is 10_2 (0 with 1 carried over). Write down 0, carry over 1.

VIIAdd the next digits (including carry-over): 0 + 1 + 1 (carry-over) = 2. In base 2, 2 is 10_2. Write down 0, carry over 1.

VIIIAdd the leftmost digits (including carry-over): 1 + 0 + 1 (carry-over) = 2. In base 2, 2 is 10_2. Write down 10.

Answer

1011_2 + 110_2 = 10001_2

Remember that in base 2, 1+1=10 (read as 'one-zero'), and 1+1+1=11 (read as 'one-one').

Example 4

4. Simplify: 3√12 + √75 - √48.

ISimplify each surd individually by finding the largest perfect square factor:

II√12 = √(4 \times 3) = √4 \times √3 = 2√3

III√75 = √(25 \times 3) = √25 \times √3 = 5√3

IV√48 = √(16 \times 3) = √16 \times √3 = 4√3

VSubstitute the simplified surds back into the expression:

VI3(2√3) + 5√3 - 4√3

VII= 6√3 + 5√3 - 4√3

VIIICombine the like surds:

9= (6 + 5 - 4)√3

10= 7√3

Answer

3√12 + √75 - √48 = 7√3

Only like surds (surds with the same number under the root sign) can be added or subtracted.

Example 5

5. Rationalise the denominator of \frac{6}{\sqrt{3}}.

ITo rationalise, multiply both the numerator and the denominator by the surd in the denominator, which is √3.

II\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}

III= \frac{6\sqrt{3}}{3}

IVSimplify the fraction:

V= 2\sqrt{3}

Answer

\frac{6}{\sqrt{3}} = 2\sqrt{3}

Remember that √a \times √a = a.

Example 6

6. Rationalise the denominator of \frac{1}{2 + \sqrt{3}}.

ITo rationalise a denominator of the form (a + √b), multiply both the numerator and the denominator by its conjugate, which is (a - √b). The conjugate of (2 + √3) is (2 - √3).

II\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}

IIIMultiply the numerators: 1 \times (2 - √3) = 2 - √3

IVMultiply the denominators using the difference of two squares formula (a+b)(a-b) = a^2 - b^2:

V(2 + √3)(2 - √3) = 2^2 - (√3)^2 = 4 - 3 = 1

VICombine the results:

VII= \frac{2 - \sqrt{3}}{1}

VIII= 2 - \sqrt{3}

Answer

\frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}

The conjugate of (a + √b) is (a - √b), and the conjugate of (a - √b) is (a + √b).

Example 7

7. Evaluate log_2 8 + log_2 4 - log_2 16.

IApply the laws of logarithms:

IIlog_2 8 = 3 (since 2^3 = 8)

IIIlog_2 4 = 2 (since 2^2 = 4)

IVlog_2 16 = 4 (since 2^4 = 16)

VSubstitute these values into the expression:

VI3 + 2 - 4

VII= 5 - 4

VIII= 1

Answer

log_2 8 + log_2 4 - log_2 16 = 1

Alternatively, use the product and quotient laws: log_2 (8 \times 4 / 16) = log_2 (32 / 16) = log_2 2 = 1.

Example 8

8. Solve for x: log_3 (x + 2) = 2.

IConvert the logarithmic equation to its exponential form. If log_b N = x, then b^x = N.

IIHere, b = 3, x = 2, and N = (x + 2).

IIISo, 3^2 = x + 2

IVCalculate 3^2:

V9 = x + 2

VIIsolate x by subtracting 2 from both sides:

VIIx = 9 - 2

VIIIx = 7

Answer

x = 7

Always check your answer by substituting it back into the original equation to ensure the argument of the logarithm is positive.

Common mistakes

✗Incorrectly carrying over or borrowing in arithmetic operations in other bases (e.g., in base 5, 3+4=12_5, not 7).
✗Treating unlike surds as like surds during addition or subtraction (e.g., √2 + √3 ≠ √5).
✗Forgetting to simplify surds to their simplest form before combining them.
✗Incorrectly applying the laws of logarithms, especially confusing log(M+N) with log M + log N (they are not equal).
✗Making errors when rationalising binomial denominators by not multiplying by the correct conjugate or misapplying the difference of squares formula.

Exam tips

★For number base conversions, show all steps clearly, especially for repeated division, to avoid errors and gain method marks.
★When simplifying surds, always look for the largest perfect square factor to reduce the number of steps.
★Memorise the laws of logarithms thoroughly, as they are essential for solving most logarithm problems.
★Practice rationalising both monomial and binomial surd denominators until it becomes second nature, as this is a common exam question.

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