Exponential and Logarithmic Functions

Exponential and Logarithmic Functions An exponential function is a function of the form $$f(x) = a^{x}$$ where $a$ is a positive real number. The graph o...

Exponential and Logarithmic Functions#

An exponential function is a function of the form $f(x) = a^{x}$ where $a$ is a positive real number. The graph of an exponential function is always increasing and approaches infinity as x approaches negative infinity.

An logarithmic function is a function of the form $f(x) = \log_a(x)$ where $a$ is a positive real number and $x$ is a positive real number. The graph of a logarithmic function is always increasing and is asymptotic to the exponential function as x approaches infinity.

Key differences:

Exponential function: The output is always positive.
Logarithmic function: The output can be both positive and negative.

Examples:

$f(x) = 2^x$ is an exponential function with base 2.
$f(x) = \log_3(x)$ is a logarithmic function with base 3.

Properties:

Exponential function:
$f(x + y) = f(x) \cdot f(y)$
$f(x)^{-1} = \frac{1}{f(x)}$
Logarithmic function:
$\log_a(x) = y \iff x = a^y$
$\log_a(x) + \log_a(y) = \log_a(xy)$
$\ln(x) = \log_e(x)$

These properties allow us to perform various operations with exponential and logarithmic functions, such as finding the exponential of a logarithm and finding the logarithm of an exponential

Exponential and Logarithmic Functions#

An exponential function is a function of the form

f(x) = a^{x}

where

a

is a positive real number. The graph of an exponential function is always increasing and approaches infinity as x approaches negative infinity.

An logarithmic function is a function of the form

f(x) = \log_a(x)

where

a

is a positive real number and

x

is a positive real number. The graph of a logarithmic function is always increasing and is asymptotic to the exponential function as x approaches infinity.

Key differences:

Exponential function: The output is always positive.

Logarithmic function: The output can be both positive and negative.

Examples:

$f(x) = 2^x$ is an exponential function with base 2.

$f(x) = \log_3(x)$ is a logarithmic function with base 3.

Properties:

Exponential function:

$f(x + y) = f(x) \cdot f(y)$

$f(x)^{-1} = \frac{1}{f(x)}$

Logarithmic function:

$\log_a(x) = y \iff x = a^y$

$\log_a(x) + \log_a(y) = \log_a(xy)$

$\ln(x) = \log_e(x)$

These properties allow us to perform various operations with exponential and logarithmic functions, such as finding the exponential of a logarithm and finding the logarithm of an exponential

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions#

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Exponential and Logarithmic Functions#