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Formulas: exponential and logarithmic functions, grade 12 (5 units)

Powers

  • A product of powers with the same base
    ax⋅ay\displaystyle a^{x}\cdot a^{y} =\displaystyle = ax+y\displaystyle a^{x+y}
  • A quotient of powers with the same base
    axay\displaystyle \frac{a^{x}}{a^{y}} =\displaystyle = ax−y\displaystyle a^{x-y}
  • A power of a power
    (ax)y\displaystyle (a^{x})^{y} =\displaystyle = ax⋅y\displaystyle a^{x\cdot y}
  • A power of a product
    (a⋅b)x\displaystyle (a\cdot b)^{x} =\displaystyle = ax⋅bx\displaystyle a^{x}\cdot b^{x}
  • A power of a quotient
    (ab)x\displaystyle \left(\frac{a}{b}\right)^{x} =\displaystyle = axbx\displaystyle \frac{a^{x}}{b^{x}}
  • A negative exponent
    a−x\displaystyle a^{-x} =\displaystyle = 1ax\displaystyle \frac{1}{a^{x}}
  • The exponent 0
    a0\displaystyle a^{0} =\displaystyle = 1\displaystyle 1a≠0\displaystyle a\neq0
  • A rational exponent
    amn\displaystyle a^{\frac{m}{n}} =\displaystyle = amn\displaystyle \sqrt[n]{a^{m}}a>0\displaystyle a>0

Logarithms (within the domain)

  • The definition of the logarithm
    log⁡ax=b  ⟺  ab=x\displaystyle \log_{a}x=b\iff a^{b}=xa>0, a≠1, x>0\displaystyle a>0,\ a\neq1,\ x>0
  • log⁡a(ab)\displaystyle \log_{a}(a^{b}) =\displaystyle = b\displaystyle b
  • alog⁡ax\displaystyle a^{\log_{a}x} =\displaystyle = x\displaystyle x
  • The logarithm of a product
    log⁡ax+log⁡ay\displaystyle \log_{a}x+\log_{a}y =\displaystyle = log⁡a(x⋅y)\displaystyle \log_{a}(x\cdot y)
  • The logarithm of a quotient
    log⁡ax−log⁡ay\displaystyle \log_{a}x-\log_{a}y =\displaystyle = log⁡a(xy)\displaystyle \log_{a}\left(\frac{x}{y}\right)
  • The logarithm of a power
    log⁡a(xb)\displaystyle \log_{a}(x^{b}) =\displaystyle = b⋅log⁡ax\displaystyle b\cdot\log_{a}x
  • Change of base
    log⁡cx\displaystyle \log_{c}x =\displaystyle = log⁡axlog⁡ac\displaystyle \frac{\log_{a}x}{\log_{a}c}
  • log⁡ab\displaystyle \log_{a}b =\displaystyle = 1log⁡ba\displaystyle \frac{1}{\log_{b}a}
  • The natural logarithm – to the base e\displaystyle e
    ln⁡x=log⁡ex\displaystyle \ln x=\log_{e}x

The exponential and the logarithmic function

  • Every exponential as a power of e\displaystyle e
    ax\displaystyle a^{x} =\displaystyle = exln⁡a\displaystyle e^{x\ln a}
  • ax\displaystyle a^x and log⁡ax\displaystyle \log_a x – inverse functions
    y=ax  ⟺  x=log⁡ay\displaystyle y=a^{x}\iff x=\log_{a}yy>0\displaystyle y>0
  • The number e\displaystyle e
    e≈2.718\displaystyle e\approx2.718

Growth and decay

  • The amount after t\displaystyle t units of time
    f(t)=f(0)⋅qt\displaystyle f(t)=f(0)\cdot q^{t}
  • The growth / decay factor – p\displaystyle p the percent per unit of time
    q=1±p100\displaystyle q=1\pm\frac{p}{100}
  • The half-life T\displaystyle T
    qT\displaystyle q^{T} =\displaystyle = 12\displaystyle \frac{1}{2}
  • The doubling time T\displaystyle T
    qT\displaystyle q^{T} =\displaystyle = 2\displaystyle 2

Derivatives

  • (ex)′\displaystyle (e^{x})^{\prime} =\displaystyle = ex\displaystyle e^{x}
  • (ax)′\displaystyle (a^{x})^{\prime} =\displaystyle = ax⋅ln⁡a\displaystyle a^{x}\cdot\ln a
  • (ln⁡x)′\displaystyle (\ln x)^{\prime} =\displaystyle = 1x\displaystyle \frac{1}{x}
  • (log⁡ax)′\displaystyle (\log_{a}x)^{\prime} =\displaystyle = 1x⋅ln⁡a\displaystyle \frac{1}{x\cdot\ln a}
  • The derivative of ef(x)\displaystyle e^{f(x)}
    [ef(x)]′\displaystyle \left[e^{f(x)}\right]^{\prime} =\displaystyle = ef(x)⋅f′(x)\displaystyle e^{f(x)}\cdot f^{\prime}(x)
  • The derivative of ln⁡(f(x))\displaystyle \ln(f(x))
    [ln⁡(f(x))]′\displaystyle \left[\ln(f(x))\right]^{\prime} =\displaystyle = f′(x)f(x)\displaystyle \frac{f^{\prime}(x)}{f(x)}
  • The derivative of a product
    [f(x)⋅g(x)]′\displaystyle \left[f(x)\cdot g(x)\right]^{\prime} =\displaystyle = f′(x)⋅g(x)+f(x)⋅g′(x)\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)
  • The derivative of a quotient
    [f(x)g(x)]′\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime} =\displaystyle = f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}
  • The derivative of a composite function
    [f(u(x))]′\displaystyle \left[f(u(x))\right]^{\prime} =\displaystyle = f′(u)⋅u′(x)\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)
  • The tangent’s equation at the point where x=a\displaystyle {x}={a}
    y−f(a)=f′(a)(x−a)\displaystyle y-f(a)=f^{\prime}(a)(x-a)

Integrals, the area and accumulation

  • ∫ex dx=ex+C\displaystyle \int e^{x}\,dx=e^{x}+C
  • ∫ax dx=axln⁡a+C\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C
  • In any interval where x≠0\displaystyle x\neq0
    ∫1x dx=ln⁡∣x∣+C\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C
  • F\displaystyle F an antiderivative of f\displaystyle f
    ∫f(mx+b) dx=1mF(mx+b)+C\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+Cm≠0\displaystyle m\neq0
  • F\displaystyle F an antiderivative of f\displaystyle f
    ∫f(u(x))⋅u′(x) dx=F(u(x))+C\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C
  • Where f(x)≠0\displaystyle f(x)\neq0
    ∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C
  • F\displaystyle F an antiderivative of f\displaystyle f
    ∫abf(x) dx=F(b)−F(a)\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)
  • The area between two graphs
    S=∫ab(f(x)−g(x))dx\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dxf(x)≥g(x)\displaystyle f(x)\geq g(x)
  • The accumulation function’s derivative
    (∫axf(t) dt)′=f(x)\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)