Powers, Exponents and Logarithms: short notes (General Aptitude)

Introduction

A power $a^n$ is repeated multiplication; an exponent is the number that says how many times; a logarithm is the exponent written as the answer. These three views of one relation, $a^x = y \iff x = \log_a y$ , are the tools for the standard numerical tasks: simplify an expression with mixed bases, solve an equation where the unknown sits in the exponent, compare two huge numbers, find a units digit, count the digits of a power, or find how many periods of compound growth reach a target.

The methods are few and mechanical. The traps are in the rules that look true but are not, such as $\ln(a + b) = \ln a + \ln b$ or $(a^b)^c = a^{(b^c)}$ , and in rounding a logarithm too early.

Prerequisites

Integer arithmetic, prime factorisation, and fractions.
For growth questions, the multiplier view of percentage change from Ratios and Percentages.

Core concepts

Term	Definition	Example
Power	$a^n = a \times a \times \cdots \times a$ ( $n$ factors) for a positive integer $n$	$2^5 = 32$
Zero exponent	$a^0 = 1$ for $a \ne 0$	$7^0 = 1$
Negative exponent	$a^{-n} = 1 / a^n$ for $a \ne 0$	$2^{-3} = 1/8$
Fractional exponent	$a^{p/q} = (\sqrt[q]{a})^p$ for $a > 0$	$27^{2/3} = 9$
Logarithm	$\log_a y = x$ means $a^x = y$ , for $a > 0$ , $a \ne 1$ , $y > 0$	$\log_2 32 = 5$
Common and natural logs	$\log_{10}$ and $\ln = \log_e$ ; these notes write $\lg$ for $\log_2$	$\log_{10} 1000 = 3$
Surd	An irrational root such as $\sqrt{2}$ or $\sqrt[3]{5}$	$\sqrt{50} = 5\sqrt{2}$

Domain convention. Fractional exponents and logarithms are used only with positive bases and positive arguments in these notes. $0^0$ is left undefined, and $(-8)^{1/3}$ is avoided because the rule $a^{p/q} = (a^{1/q})^p$ breaks for negative $a$ with some representations of the same fraction.

The laws

For $a, b > 0$ and real $m, n$ :

Law	Statement
Product	$a^m a^n = a^{m+n}$
Quotient	$a^m / a^n = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$
Power of a product	$(ab)^n = a^n b^n$
Log of a product	$\log_c (xy) = \log_c x + \log_c y$
Log of a quotient	$\log_c (x/y) = \log_c x - \log_c y$
Log of a power	$\log_c (x^k) = k \log_c x$
Change of base	$\log_a y = \log_c y / \log_c a$ for any valid base $c$
Inverse pair	$a^{\log_a y} = y$ and $\log_a (a^x) = x$
Reciprocal	$\log_a b = 1 / \log_b a$

Each log law is an exponent law read backwards: if $x = c^u$ and $y = c^v$ , then $xy = c^{u+v}$ , so $\log_c(xy) = u + v$ .

Decision procedure

Same base first. Rewrite every number as a power of a common prime: $4 = 2^2$ , $8 = 2^3$ , $27 = 3^3$ , $125 = 5^3$ . Then exponents simply add and subtract.
Unknown in the exponent. Bring both sides to the same base and equate exponents; if that is impossible, take logarithms of both sides.
Comparing two powers. Make the exponents equal ( $2^{300} = 8^{100}$ , $3^{200} = 9^{100}$ ) or the bases equal; failing both, compare $n \ln a$ .
Units digit. For positive integer exponents, units digits of powers repeat with period 1, 2 or 4. Reduce the exponent modulo 4 (use 4 when the remainder is 0).
Number of digits. A positive integer $N$ has $\lfloor \log_{10} N \rfloor + 1$ digits.
Growth to a target. For $m>1$ , $k\ge1$ and a nonnegative integer number of periods $n$ , the least $n$ with $m^n\ge k$ is $\lceil\ln k/\ln m\rceil$ . Confirm with the powers on either side. If $k\le1$ , the target is already reached at $n=0$ . For $m=1$ or decay $0<m<1$ , handle reachability separately; do not apply this ceiling blindly.
Logarithmic equations. Combine logs into one, convert to exponential form, solve, then reject any root that makes an argument non-positive.

Mathematical explanation

Why units digits cycle. Handle exponent 0 first: $a^0=1$ for $a\ne0$ ; $0^0$ is undefined under this note's convention. For positive integer exponents: The units digit of $a^n$ depends only on the units digit of $a$ and is obtained by multiplying the previous units digit by it. There are at most ten digits, so the sequence repeats; checking the ten cases shows every cycle length divides 4:

Last digit of $a$	Units digits of $a^1, a^2, a^3, a^4$	Period
0, 1, 5, 6	the digit itself	1
4	4, 6, 4, 6	2
9	9, 1, 9, 1	2
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4

Digit count. If $N$ has $d$ digits then $10^{d-1} \le N < 10^d$ , so $d - 1 \le \log_{10} N < d$ , which gives $d = \lfloor \log_{10} N \rfloor + 1$ .

Change of base. Let $x = \log_a y$ , so $a^x = y$ . Taking $\log_c$ of both sides, $x \log_c a = \log_c y$ , which is the change-of-base rule.

Growth. For $m>1$ , $k\ge1$ , solving $m^n=k$ gives $n=\ln k/\ln m$ ; the first nonnegative whole period at or above the target is its ceiling. For $k\le1$ , use zero periods; constant or decaying multipliers need a separate reachability check.

Worked examples

Example 1 (same base). Simplify $\frac{2^5 \times 4^3}{8^2}$ .

$4^3 = (2^2)^3 = 2^6$ and $8^2 = (2^3)^2 = 2^6$ .
$\frac{2^5 \times 2^6}{2^6} = 2^5 = 32$ .

Example 2 (unknown in the exponent). Solve $3^{2x - 1} = 243$ .

$243 = 3^5$ , so $2x - 1 = 5$ and $x = 3$ .

Example 3 (fractional and negative exponents). $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$ , and $16^{-3/4} = 1 / (\sqrt[4]{16})^3 = 1 / 2^3 = 1/8$ .

Example 4 (comparison). Which is larger, $2^{300}$ or $3^{200}$ ?

$2^{300} = (2^3)^{100} = 8^{100}$ and $3^{200} = (3^2)^{100} = 9^{100}$ .
Equal exponents, larger base wins: $3^{200}$ is larger.

Example 5 (units digit). Find the units digit of $7^{123}$ .

The cycle for 7 is 7, 9, 3, 1 (period 4).
$123 = 4 \times 30 + 3$ , so take the 3rd term of the cycle: the units digit is 3.

Example 6 (number of digits). How many digits does $2^{100}$ have? Use $\log_{10} 2 \approx 0.30103$ .

$\log_{10} 2^{100} = 100 \times 0.30103 = 30.103$ .
Digits: $\lfloor 30.103 \rfloor + 1 = 31$ .

Example 7 (log arithmetic). Evaluate $\log_2 8 + \log_3 81 - \log_5 125$ .

$\log_2 8 = 3$ , $\log_3 81 = 4$ , $\log_5 125 = 3$ , so the value is $3 + 4 - 3 = 4$ .

Example 8 (building a log from given values). Given $\log_{10} 2 = 0.3010$ and $\log_{10} 3 = 0.4771$ , find $\log_{10} 72$ .

$72 = 2^3 \times 3^2$ , so $\log_{10} 72 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572$ .
With more decimal places the true value is $1.85733$ to five places; the four-place inputs give the answer to within 0.0002.

Example 9 (change of base). $\log_4 32 = \frac{\log_2 32}{\log_2 4} = \frac{5}{2}$ .

Example 10 (logarithmic equation with a rejected root). Solve $\log_{10} x + \log_{10}(x - 3) = 1$ .

Combine: $\log_{10}(x(x - 3)) = 1$ , so $x^2 - 3x = 10$ , that is $x^2 - 3x - 10 = 0$ , giving $x = 5$ or $x = -2$ .
$x = -2$ makes $\log_{10} x$ undefined, so the only solution is $x = 5$ .

Example 11 (growth to a target). Money grows at 8% per year, compounded annually. After how many whole years has it first at least doubled?

$n = \ln 2 / \ln 1.08 \approx 9.006$ , so the ceiling is 10.
Check by powers: $1.08^9 \approx 1.999$ (just short of 2) and $1.08^{10} \approx 2.159$ . The answer is 10 years. Rounding the logarithm to 9 gives the wrong option.

Example 12 (surds). $\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ , and $\frac{1}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} + \sqrt{2}}{3 - 2} = \sqrt{3} + \sqrt{2}$ (multiply by the conjugate).

Important properties and observations

Common misconceptions. $\ln(a + b) \ne \ln a + \ln b$ ; $(a + b)^2 \ne a^2 + b^2$ ; $a^{m} a^{n} \ne a^{mn}$ ; $\log_a b \cdot \log_b a = 1$ (not $\log_a (ab)$ ).
Tower convention. $a^{b^c}$ means $a^{(b^c)}$ , which differs from $(a^b)^c = a^{bc}$ : $2^{3^2} = 2^9 = 512$ but $(2^3)^2 = 64$ .
Monotonicity. For base $a > 1$ , $\log_a$ is increasing, so inequalities survive taking logs; for $0 < a < 1$ they reverse.
Logs of numbers near powers of 10. $\log_{10}$ of a number between 1 and 10 lies between 0 and 1; multiplying the number by $10^k$ adds $k$ to its base-10 logarithm (a rightward decimal shift by $k$ places for positive $k$ ); division by $10^k$ subtracts $k$ .
Useful values. $\log_{10} 2 \approx 0.3010$ , $\log_{10} 3 \approx 0.4771$ , $\log_{10} 5 = 1 - \log_{10} 2 \approx 0.6990$ , $\log_{10} 7 \approx 0.8451$ , $2^{10} = 1024 \approx 10^3$ .
EDGE CASE. The base of a logarithm can never be 1, and the argument must be positive; candidates that break this are rejected (Example 10).
Related material. The "rule of 70" estimates the doubling time at $r\%$ growth as about $70 / r$ periods (from $\ln 2 \approx 0.693$ ); at 8% it suggests about 8.75 years, which shows why an estimate must be confirmed by powers when the question asks for whole periods.

GATE examination perspective

Base-rewriting questions. Expressions with 4, 8, 16, 32 or 9, 27, 81 collapse to one base when rewritten (Example 1).
Exponent-equation traps. After equating exponents, solve the linear equation carefully; a sign slip in $2x - 1$ is a typical distractor (Example 2).
Comparison traps. The larger exponent does not decide by itself; equalise exponents or compare $n \ln a$ (Example 4).
Units-digit questions. Handle exponent 0 separately; for positive integer exponents, reduce the exponent modulo 4, and use the 4th term when the remainder is 0.
Digit-count questions. Use $\lfloor \log_{10} N \rfloor + 1$ , and remember that $10^k$ itself has $k + 1$ digits.
Off-by-one in growth. Ceil the exact quotient and verify with the neighbouring powers (Example 11).
Domain traps. Logarithmic equations often produce an extraneous root (Example 10).

Summary

$a^x = y \iff x = \log_a y$ for $a > 0$ , $a \ne 1$ , $y > 0$ .
Exponent laws: add on multiplying, subtract on dividing, multiply for a power of a power.
Log laws are the exponent laws read backwards; change of base is $\log_a y = \log_c y / \log_c a$ .
Rewrite numbers in a common prime base before anything else.
For positive integer exponents, units digits of powers cycle with period 1, 2 or 4.
A positive integer $N$ has $\lfloor \log_{10} N \rfloor + 1$ digits.
For $m>1,k\ge1$ and integer $n\ge0$ , the least $n$ with $m^n\ge k$ is $\lceil\ln k/\ln m\rceil$ . If $k\le1$ , it is zero; check other multipliers separately.
Reject solutions that make a logarithm's argument non-positive.

Ratios and Percentages: prerequisite. Multipliers for percentage growth.
Asymptotic Notation: related. Logarithms and powers in growth-rate comparisons.
Number and Letter Series: related. Geometric sequences are powers of the common ratio.

Powers, Exponents and Logarithms

Introduction

Prerequisites

Core concepts

The laws

Decision procedure

Mathematical explanation

Worked examples

Important properties and observations

GATE examination perspective

Summary

Sources

Next steps

Introduction

Prerequisites

Core concepts

The laws

Decision procedure

Mathematical explanation

Worked examples

Important properties and observations

GATE examination perspective

Summary

Related topics

Sources

Next steps