Logarithms: Properties and Change of Base | NIMCET

Quick answer: $\log_a x = y$ means $a^y = x$ , for $a > 0$ , $a \ne 1$ and $x > 0$ . The main rules are $\log_a(mn) = \log_a m + \log_a n$ , $\log_a\frac{m}{n} = \log_a m - \log_a n$ and $\log_a m^k = k\log_a m$ . The change of base formula is $\log_a x = \frac{\log_b x}{\log_b a}$ , which gives $\log_a b \cdot \log_b a = 1$ and $a^{\log_a x} = x$ .

Logarithms are the inverse of the exponential functions in indices and surds.

Definition and conditions

$\log_a x = y \iff a^y = x$

The logarithm is defined only when the base $a > 0$ , $a \ne 1$ and the argument $x > 0$ .

Special value	Result
$\log_a 1$	0
$\log_a a$	1
$a^{\log_a x}$	$x$
$\log_a a^x$	$x$

Laws of logarithms

Law	Statement
Product	$\log_a(mn) = \log_a m + \log_a n$
Quotient	$\log_a\frac{m}{n} = \log_a m - \log_a n$
Power	$\log_a m^k = k\log_a m$
Base power	$\log_{a^k} m = \frac{1}{k}\log_a m$
Change of base	$\log_a x = \frac{\log_b x}{\log_b a}$
Reciprocal	$\log_a b = \frac{1}{\log_b a}$
Chain	$\log_a b \cdot \log_b c \cdot \log_c d = \log_a d$
Swap	$a^{\log_b c} = c^{\log_b a}$

Common values (base 10)

Number	$\log_{10}$ (approx.)
2	0.3010
3	0.4771
5	0.6990
7	0.8451

Number of digits: a positive integer N has $\lfloor\log_{10}N\rfloor + 1$ digits. For example, $2^{10} = 1024$ , and $\log_{10}2^{10} \approx 3.01$ , giving 4 digits.

Logarithmic inequalities

Base	$\log_a x > \log_a y$ means
$a > 1$	$x > y > 0$
$0 < a < 1$	$0 < x < y$ (inequality reverses)

Worked examples

Example 1: simplify

Find $\log_2 8 + \log_3 \frac{1}{9} - \log_5 1$ .

$3 + (-2) - 0 = 1$ .

Example 2: chain rule

Find $\log_3 4 \cdot \log_4 5 \cdot \log_5 9$ .

The chain gives $\log_3 9 = 2$ .

Example 3: logarithmic equation

Solve $\log_2(x + 1) + \log_2(x - 1) = 3$ .

Step 1: $\log_2(x^2 - 1) = 3$ , so $x^2 - 1 = 8$ and $x = \pm 3$ .

Step 2: Both arguments must be positive, so $x > 1$ . Answer: $x = 3$ .

Example 4: number of digits

How many digits does $3^{20}$ have?

$\log_{10}3^{20} = 20 \times 0.4771 = 9.542$ , so it has 10 digits.

Example 5: inequality

Solve $\log_{\frac{1}{2}}(x - 2) > -1$ .

The base is less than 1, so the inequality reverses: $0 < x - 2 < 2$ , giving $2 < x < 4$ .

Summary table

Idea	Key result
Definition	$\log_a x = y \iff a^y = x$
Change of base	$\log_a x = \frac{\log_b x}{\log_b a}$
Reciprocal	$\log_a b \cdot \log_b a = 1$
Domain	Base positive and not 1; argument positive
Base below 1	Inequalities reverse

Exam tips

Always check the domain of every log term after solving; reject roots that make an argument non-positive.
For products of logs with chained bases, cancel them like fractions.
Use $\log_{10}2 \approx 0.3010$ and $\log_{10}3 \approx 0.4771$ for digit-count questions.

Continue with quadratic equations, or see the NIMCET Algebra chapter.

Common mistakes

Writing $\log(m + n) = \log m + \log n$ .
Writing $\frac{\log m}{\log n} = \log m - \log n$ .
Forgetting to reverse the inequality when the base is between 0 and 1.
Keeping roots that make the argument of a log negative.

Frequently Asked Questions

What is the change of base formula?

log_a x = log_b x / log_b a, for any valid base b.

What is log_a b × log_b a?

It equals 1.

When is a logarithm defined?

log_a x is defined when a > 0, a ≠ 1 and x > 0.

How do you find the number of digits in a large number?

A positive integer N has ⌊log₁₀ N⌋ + 1 digits.

What happens to a log inequality when the base is less than 1?

The inequality reverses, because the logarithm is decreasing for such a base.