MAT.126
5.5
Bases Other Than e and Applications

Bases Other than e

Definition of Exponential Function to Base a

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If a is a positive real number (a<>1) and x is any real number, then the exponential function to the base a is denoted by a^x and is defined bya^x = e^{(ln a)x}If a=1, then y=1^x=1 is a constant function.

Laws of Exponents

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a^0 = 1a^x a^y = a^{x+y}a^x / a^y = a^{x-y}(a^x)^y = a^{xy}

Definition of Logarithmic Function to Base a

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If a is a positive real number (a<>1) and x is any positive real number, then the logarithmic function to the base a is denoted by log_a x and is defined aslog_a x = (1/ln a) ln x

Logarithmic Properties

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log_a 1 = 0log_a xy = log_a x + log_a ylog_a x/y = log_a x - log_a ylog_a x^n = n log_a x

Inverse Function Properties

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y = a^x if and only if x = log_a ya^{log_a x} = x, for x > 0log_a a^x = x, for all x

Common Logarithmic Function

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The logarithm with base 10.

Applications of Exponential Functions

Compound Interest

Continuously Compounded Interest

Logisitic Growth

Differentiation and Integration

Derivatives for Bases Other Than e

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Let a be a positive real number (a<>1) and let u be a differentiable function of x. d---[a^x] = (ln a) a^x dx d du---[a^u] = (ln a) a^u ---- dx dx d 1---[log_a x] = -------- dx (ln a) x d 1 du---[log_a u] = -------- --- dx (ln a) u dx

Integration

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When confronted with an integral of the formS a^x dxthere are two choices.One is to convert the exponential expression with base a to an equivalent exponential expression with base e. That is, considerS e^{(ln a)x} dxremembering that ln a is a constant.The second option is to use the following integration formula,S a^x dx = (1/ln a) a^x + C

A comparison of 4 rules for differentiation

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Constant Rule d/dx[e^e] = 0Power Rule d/dx[x^e] = e x^{e-1}Exponential Rule d/dx[e^x] = e^xLogarithmic differentiation d/dx[x^x] = x^x (1 + ln x)