Powers, Roots and Logarithms

Rules for exponents, square roots and logarithms: multiplying and dividing powers, zero, negative and rational exponents, the definition and properties of logarithms and the change-of-base formula. Quick formulas for percentages are included as well.

24 formulas · Grades 5–11

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All 24 formulas on this page
  1. Finding 1% of a number
  2. Finding p% of a number A
  3. Finding a number from its percentage
  4. Multiplying powers
  5. Dividing powers
  6. Power of a power
  7. Power of a product
  8. Zero exponent
  9. Properties of a square root (multiplication)
  10. Properties of a square root (division)
  11. Square of a root
  12. Root of a square
  13. Negative exponent
  14. Rational exponent
  15. Definition of a logarithm
  16. Fundamental logarithmic identity
  17. Properties of logarithms (sum)
  18. Properties of logarithms (difference)
  19. Taking an exponent out of a logarithm
  20. Change-of-base formula
  21. Logarithm of 1
  22. Logarithm of a number equal to the base
  23. Logarithm with the base raised to a power
  24. Reciprocal logarithm bases

Finding 1% of a number

Grade 5
A100\frac{A}{100}

One hundredth of a given number A.

Legend:
  • A — Number A

Finding p% of a number A

Grade 5
B=A⋅p100B = A \cdot \frac{p}{100}

The value of a given percentage p of a number A.

Legend:
  • A — Number A
  • p — Percentage p

Finding a number from its percentage

Grade 5
A=B⋅100pA = B \cdot \frac{100}{p}

The number A, given that B is p% of it.

Legend:
  • B — Known part (B)
  • p — Percentage p

Multiplying powers

Grade 7
am⋅an=am+na^m \cdot a^n = a^{m+n}

When multiplying powers with the same base, the exponents are added.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Dividing powers

Grade 7
am:an=am−na^m : a^n = a^{m-n}

When dividing powers with the same base, the exponents are subtracted.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Power of a power

Grade 7
(am)n=am⋅n(a^m)^n = a^{m \cdot n}

When raising a power to a power, the exponents are multiplied.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Power of a product

Grade 7
(a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n

Each factor is raised to the given power.

Legend:
  • a — Factor a
  • b — Factor b
  • n — Exponent n

Zero exponent

Grade 7
a0=1(a≠0)a^0 = 1 \quad (a \neq 0)

Any nonzero number raised to the power of 0 equals 1.

Legend:
  • a — Base a (a≠0)

Properties of a square root (multiplication)

Grade 8
a⋅b=a⋅b(a,b≥0)\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \quad (a, b \ge 0)

The root of a product of non-negative numbers equals the product of their roots.

Legend:
  • a — Number a (a≥0)
  • b — Number b (b≥0)

Properties of a square root (division)

Grade 8
ab=ab(a≥0,b>0)\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (a \ge 0, b > 0)

The root of a quotient equals the quotient of the roots of the numerator and denominator.

Legend:
  • a — Number a (a≥0)
  • b — Number b (b>0)

Square of a root

Grade 8
(a)2=a(a≥0)(\sqrt{a})^2 = a \quad (a \ge 0)

Squaring the arithmetic square root.

Legend:
  • a — Number a (a≥0)

Root of a square

Grade 8
a2=∣a∣\sqrt{a^2} = |a|

The square root of the square of a number equals its absolute value.

Legend:
  • a — Number a

Negative exponent

Grade 8
a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \quad (a \neq 0)

Replacing a negative exponent with the reciprocal value.

Legend:
  • a — Base a (a≠0)
  • n — Exponent n

Rational exponent

Grade 10
amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Expressing a power with a fractional exponent as a root.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Root index n (n≠0)

Definition of a logarithm

Grade 10
log⁡ab=c  ⟺  ac=b(a>0,a≠1,b>0)\log_a b = c \iff a^c = b \quad (a>0, a \neq 1, b>0)

Finds the exponent c to which base a must be raised to get b.

Legend:
  • a — Base a (a>0, a≠1)
  • b — Argument b (b>0)

Fundamental logarithmic identity

Grade 10
alog⁡ab=ba^{\log_a b} = b

Raising base a to the power log_a(b) gives the argument b.

Legend:
  • b — Number b (b>0)

Properties of logarithms (sum)

Grade 10
log⁡ax+log⁡ay=log⁡a(x⋅y)\log_a x + \log_a y = \log_a (x \cdot y)

The sum of logarithms equals the logarithm of the product of the arguments.

Legend:
  • a — Base a
  • x — Argument x
  • y — Argument y

Properties of logarithms (difference)

Grade 10
log⁡ax−log⁡ay=log⁡a(xy)\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right)

The difference of logarithms equals the logarithm of the quotient of the arguments.

Legend:
  • a — Base a
  • x — Argument x
  • y — Argument y

Taking an exponent out of a logarithm

Grade 10
log⁡a(xk)=k⋅log⁡ax\log_a (x^k) = k \cdot \log_a x

The exponent k is moved out as a factor in front of the logarithm.

Legend:
  • a — Base a
  • x — Argument x
  • k — Exponent k

Change-of-base formula

Grade 10
log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}

Converting a logarithm with base a to base c.

Legend:
  • a — Old base a
  • b — Argument b
  • c — New base c

Logarithm of 1

Grade 10
log⁡a1=0\log_a 1 = 0

The logarithm of 1 to any base is always 0.

Legend:
  • a — Base a (a>0, a≠1)

Logarithm of a number equal to the base

Grade 10
log⁡aa=1\log_a a = 1

The logarithm of a number to an equal base is always 1.

Legend:
  • a — Base a (a>0, a≠1)

Logarithm with the base raised to a power

Grade 11
log⁡anx=1nlog⁡ax\log_{a^n} x = \frac{1}{n}\log_a x

Lets you move the exponent of the base out from under the logarithm (for n ≠ 0).

Legend:
  • n — Exponent n
  • a — Base a (a>0, a≠1)
  • x — Argument x

Reciprocal logarithm bases

Grade 11
log⁡ax=1log⁡xa\log_a x = \frac{1}{\log_x a}

Relates the logarithm with base a to the logarithm with the reciprocal base.

Legend:
  • a — Base a (a>0, a≠1)
  • x — Argument x

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