Laws of Exponential and Logarithmic Functions

Laws of Exponential and Logarithmic Functions#

Laws, Properties, and Derivatives#

Let $a$ and $b$ be positive numbers and $x$ and $y$ be real numbers. Let $m$ and $n$ be positive numbers.

Exponential

Addition of Exponents Law

$b^{x} b^{y} = b^{x + y}$
Difference of Exponents Law

$\frac{b^{x}}{b^{y}} = b^{x - y}$
Exponentiation Law

${(b^{x})}^{y} = b^{x y}$
Product Distribution Law

$(a b)^{x} = a^{x} b^{x}$
Fractional Distribution Law

${(\frac{a}{b})}^{x} = \frac{a^{x}}{b^{x}}$

Logarithmic

Logarithmic Addition Law

$\log_{b} (m n) = \log_{b} (m) + \log_{b} (n)$
Logarithmic Subtraction Law

$\log_{b} (\frac{m}{n}) = \log_{b} (m) - \log_{b} (n)$
Logarithm of a Power

$\log_{b} (m^{n}) = n \log_{b} (m)$
Logarithm of 1

$\log_{b} (1) = 0 & \ln (1) = 0$
Logarithm of the Base

$\log_{b} (b) = 1 & \ln (e) = 1$

Cancellation Properties of $e^{x}$ and $\ln (x)$

For all $x > 0$

$e^{\ln (x)} = x$

For all $x$

$\ln (e^{x}) = x$

Derivatives of $e^{x}$ and $\ln (x)$

Derivative of $e^{x}$

$\frac{d}{d x} e^{x} = e^{x}$
Derivative of $e^{f (x)}$

$\frac{d}{d x} e^{f (x)} = e^{f (x)} f^{'} (x)$

Derivative of $\ln (x)$

$\frac{d}{d x} \ln (x) = \frac{1}{x}$
Derivative of $\ln (f (x))$

$\frac{d}{d x} \ln (f (x)) = \frac{1}{f (x)} f^{'} (x)$

Example 1#

Solve for unknown value in the exponent

Find all values of $x$ such that $4^{x - x^{2}} = \frac{1}{16^{x}}$ .

Example 2#

Solve for unknown value in the exponent

Find all values of $x$ such that:

2^{2 x} - 40 \cdot 2^{x} + 256 = 0.

Example 3#

Solve for unknown value in the exponent

Find all values of $t$ such that $\frac{360}{1 + 9 e^{- 2 t}} = 90$ .

Example 4#

Expand using properties of logarithms

Expand the following expression:

\ln (\frac{\sqrt[3]{(x + 1)^{2}} \cdot e^{5 x}}{x}) .

Example 5#

Equation of the tangent line

Find the tangent line to $y = \frac{e^{27 x}}{x^{9}}$ at the point $(1, e^{27})$ .

Example 6#

Marginal revenue function

Suppose the unit selling price $p (x)$ and the quantity supplied $x$ of a certain product is given by

p (x) = x^{3} e^{5 x} + 12.

Find the marginal revenue function $R^{'} (x)$ .

Example 7#

Derivative of a logarithmic function

Compute the derivative of $f (x) = \ln (\frac{\sqrt{6 x + 1}}{5 x})$ .

Example 8#

Implicit differentiation

Let $\ln (x y) + y^{7} = x^{3} + 2 x$ . Find $\frac{d y}{d x}$ .

Laws of Exponential and Logarithmic Functions

Contents

Laws of Exponential and Logarithmic Functions#

Laws, Properties, and Derivatives#

Example 1#

Example 2#

Example 3#

Example 4#

Example 5#

Example 6#

Example 7#

Example 8#