Exponents And Logarithms

Exponential and logarithmic functions are inverse operations. Exponential growth and decay models, along with logarithmic differentiation, rely heavily on mastering exponent rules and logarithm properties.

Quick Reference

Property / Rule	Expression	Notes
Product of Powers	$a^{x} a^{y} = a^{x + y}$	Add exponents with same base
Power of a Power	$(a^{x})^{y} = a^{x y}$	Multiply exponents
Logarithm Definition	$y = \log_{a} x ⟺ a^{y} = x$	$a > 0, a \neq 1, x > 0$
Product Rule for Logs	$\log_{a} (x y) = \log_{a} x + \log_{a} y$	Log of product is sum of logs
Power Rule for Logs	$\log_{a} (x^{r}) = r \log_{a} x$	Exponent pulls out as multiplier
Change of Base	$\log_{a} x = \frac{\ln x}{\ln a}$	Conversion to natural logarithm

Laws of Exponents

Let a and b be positive numbers and x and y be real numbers. Then

$a^{x + y} = a^{x} a^{y}$
$a^{x - y} = \frac{a^{x}}{a^{y}}$
$(a^{x})^{y} = a^{x y}$
$(a b)^{x} = a^{x} b^{x}$
${(\frac{a}{b})}^{x} = \frac{a^{x}}{b^{x}}$
$a^{0} = 1$
$a^{- x} = \frac{1}{a^{x}}$

Logarithms

Let $a > 0$ and $a \neq 1$ . The logarithm of x to base a is denoted by $\log_{a} x$ and is defined as:

y = \log_{a} x ⟺ a^{y} = x (x > 0)

When $a = e$ ( $e \approx 2.71828$ ), we call $\log_{e} x$ the natural logarithm of x and write $\ln x$ :

y = \ln x ⟺ e^{y} = x (x > 0)

Properties of Logarithms

Let x and y be positive numbers and r be a real number. Then

$\log_{a} (x y) = \log_{a} x + \log_{a} y$
$\log_{a} (\frac{x}{y}) = \log_{a} x - \log_{a} y$
$\log_{a} (x^{r}) = r \log_{a} x$
$\log_{a} 1 = 0$
$\log_{a} a = 1$
$a^{\log_{a} x} = x$ and $\log_{a} (a^{x}) = x$

Change of Base Formula

For any positive base $a \neq 1$ and positive number $x$ :

\log_{a} x = \frac{\ln x}{\ln a} = \frac{\log_{10} x}{\log_{10} a}

Quick Reference

Laws of Exponents

Logarithms

Properties of Logarithms

Change of Base Formula

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