Mathematics

Exponents, Radicals, and Logarithms

Properties of exponents and radicals, operations with radicals (simplification, rationalization), and properties of logarithms including change of base.

Property
Example
1.a0=1\mathbf{1.}\quad a^0 = 1
50=15^0 = 1
2.a1=a\mathbf{2.}\quad a^1 = a
51=55^1 = 5
3.a−n=1an\mathbf{3.}\quad a^{-n} = \dfrac{1}{a^n}
5−2=152=1255^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}
4.amn=amn\mathbf{4.}\quad a^{\frac{m}{n}} = \sqrt[n]{a^m}
523=5235^{\frac{2}{3}} = \sqrt[3]{5^2}
5.am⋅an=am+n\mathbf{5.}\quad a^m \cdot a^n = a^{m+n}
52⋅53=52+3=555^2 \cdot 5^3 = 5^{2+3} = 5^5
6.aman=am−n\mathbf{6.}\quad \dfrac{a^m}{a^n} = a^{m-n}
5352=53−2=51\dfrac{5^3}{5^2} = 5^{3-2} = 5^1
7.(am)n=am⋅n\mathbf{7.}\quad (a^m)^n = a^{m \cdot n}
(52)3=52⋅3=56(5^2)^3 = 5^{2 \cdot 3} = 5^6
8.(a⋅b)n=an⋅bn\mathbf{8.}\quad (a \cdot b)^n = a^n \cdot b^n
(5⋅3)2=52⋅32(5 \cdot 3)^2 = 5^2 \cdot 3^2
9.(ab)n=anbn\mathbf{9.}\quad \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}
(53)2=5232\left(\dfrac{5}{3}\right)^2 = \dfrac{5^2}{3^2}
Note for property 3: Applied to a fraction it results in:
(ab)−n=(ba)n=bnan\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n = \dfrac{b^n}{a^n}
Property
Example
1.a⋅bn=an⋅bn\mathbf{1.}\quad \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}
5⋅33=53⋅33\sqrt[3]{5 \cdot 3} = \sqrt[3]{5} \cdot \sqrt[3]{3}
2.abn=anbn\mathbf{2.}\quad \sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}
533=5333\sqrt[3]{\dfrac{5}{3}} = \dfrac{\sqrt[3]{5}}{\sqrt[3]{3}}
3.(an)m=amn\mathbf{3.}\quad (\sqrt[n]{a})^m = \sqrt[n]{a^m}
(53)2=523(\sqrt[3]{5})^2 = \sqrt[3]{5^2}
4.anm=am⋅n\mathbf{4.}\quad \sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}
53=56\sqrt{\sqrt[3]{5}} = \sqrt[6]{5}
5.amn=am⋅kn⋅k\mathbf{5.}\quad \sqrt[n]{a^m} = \sqrt[n \cdot k]{a^{m \cdot k}}
53=564\sqrt{5^3} = \sqrt[4]{5^6}
6.amn=amn\mathbf{6.}\quad \sqrt[n]{a^m} = a^{\frac{m}{n}}
523=523\sqrt[3]{5^2} = 5^{\frac{2}{3}}
Sign of a radical:
  • If the index is odd, the radical has the same sign as the radicand:
−83=−83=−283=83=2\sqrt[3]{-8} = -\sqrt[3]{8} = -2 \qquad \sqrt[3]{8} = \sqrt[3]{8} = 2
  • If the index is even and the radicand is positive, there are two solutions: positive and negative.
164=±2\sqrt[4]{16} = \pm 2
  • If the index is even and the radicand is negative, there is no real solution.
−16∉R\sqrt{-16} \notin \mathbb{R}
Simplifying a radical (extracting factors): Find the prime factorization of the radicand. Ex.:
648.0003=26⋅34⋅533=22⋅3⋅5⋅33\sqrt[3]{648{.}000} = \sqrt[3]{2^6 \cdot 3^4 \cdot 5^3} = 2^2 \cdot 3 \cdot 5 \cdot \sqrt[3]{3}
Reducing radicals to a common index: The common index is the LCM of the indices. Use property 5. Ex.:
323⋅5⋅74=3812⋅5612⋅7312=38⋅56⋅7312\sqrt[3]{3^2} \cdot \sqrt{5} \cdot \sqrt[4]{7} = \sqrt[12]{3^{8}} \cdot \sqrt[12]{5^{6}} \cdot \sqrt[12]{7^{3}} = \sqrt[12]{3^8 \cdot 5^6 \cdot 7^3}
Multiplying and dividing radicals: Reduce to a common index and use properties 1 and 2. Ex.:
323⋅5154=3812⋅561215312=38⋅5633⋅5312=35⋅5312\dfrac{\sqrt[3]{3^2} \cdot \sqrt{5}}{\sqrt[4]{15}} = \dfrac{\sqrt[12]{3^8} \cdot \sqrt[12]{5^6}}{\sqrt[12]{15^3}} = \sqrt[12]{\dfrac{3^8 \cdot 5^6}{3^3 \cdot 5^3}} = \sqrt[12]{3^5 \cdot 5^3}
Adding and subtracting like radicals: Ex.:
27+48−34=33+43−123=1323\sqrt{27} + \sqrt{48} - \sqrt{\dfrac{3}{4}} = 3\sqrt{3} + 4\sqrt{3} - \dfrac{1}{2}\sqrt{3} = \dfrac{13}{2}\sqrt{3}
Rationalizing the denominator: Removing roots from the denominator:
  • If the denominator is a square root:
13=13⋅33=33\dfrac{1}{\sqrt{3}} = \dfrac{1}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}
  • If the denominator is an nth root:
1325=1325⋅335335=335355=3353\dfrac{1}{\sqrt[5]{3^2}} = \dfrac{1}{\sqrt[5]{3^2}} \cdot \dfrac{\sqrt[5]{3^3}}{\sqrt[5]{3^3}} = \dfrac{\sqrt[5]{3^3}}{\sqrt[5]{3^5}} = \dfrac{\sqrt[5]{3^3}}{3}
  • If the denominator is a binomial:
1(5−3)=1(5−3)⋅(5+3)(5+3)=(5+3)(5)2−(3)2=(5+3)2\dfrac{1}{(\sqrt{5}-\sqrt{3})} = \dfrac{1}{(\sqrt{5}-\sqrt{3})} \cdot \dfrac{(\sqrt{5}+\sqrt{3})}{(\sqrt{5}+\sqrt{3})} = \dfrac{(\sqrt{5}+\sqrt{3})}{(\sqrt{5})^2-(\sqrt{3})^2} = \dfrac{(\sqrt{5}+\sqrt{3})}{2}
Definition:
log⁡ba=x⇒bx=a\log_b a = x \Rightarrow b^x = a
Special logarithms:
Common logarithm
Natural logarithm
log⁡10a=log⁡a\log_{10} a = \log a
log⁡ea=ln⁡a\log_e a = \ln a
Change of base formula:
log⁡ca=log⁡balog⁡bc\log_c a = \dfrac{\log_b a}{\log_b c}
Properties of Logarithms
Property
Example
log⁡bb=1\log_b b = 1
log⁡22=1;log⁡10=1\log_2 2 = 1 \quad ; \quad \log 10 = 1
log⁡b1=0\log_b 1 = 0
log⁡1=0\log 1 = 0
log⁡a+log⁡b=log⁡(a⋅b)\log a + \log b = \log(a \cdot b)
log⁡2+log⁡3=log⁡(2⋅3)\log 2 + \log 3 = \log(2 \cdot 3)
log⁡a−log⁡b=log⁡ ⁣(ab)\log a - \log b = \log\!\left(\dfrac{a}{b}\right)
log⁡18−log⁡2=log⁡ ⁣(182)\log 18 - \log 2 = \log\!\left(\dfrac{18}{2}\right)
log⁡ab=b⋅log⁡a\log a^b = b \cdot \log a
log⁡23=3⋅log⁡2\log 2^3 = 3 \cdot \log 2
log⁡an=log⁡a1n=1n⋅log⁡a\log \sqrt[n]{a} = \log a^{\frac{1}{n}} = \dfrac{1}{n} \cdot \log a
log⁡3=log⁡312=12⋅log⁡3\log \sqrt{3} = \log 3^{\frac{1}{2}} = \dfrac{1}{2} \cdot \log 3