We can transform graphs of exponential functions and logarithmic functions in the same way we transform any other functions. Let's review some of the more important transformations.

Notation	Transformation	Examples
$f (x) \to f (x - h)$	Horizontal Translation Shifts the graph left if $h < 0,$ or right if $h > 0$	$g (x) = 4^{x - 3},$ shifts the graph of $f (x) = 4^{x}$ to the right by $3$ units.
$f (x) \to f (x - h)$		$g (x) = 4^{x + 2},$ shifts the graph of $f (x) = 4^{x}$ to the left by $2$ units.
$f (x) \to f (x) + k$	Vertical Translation Shifts the graph up if $k > 0,$ or down if $k < 0$	$g (x) = lo g_{2} (x) + 5,$ shifts the graph of $f (x) = lo g_{2} (x)$ up by $5$ units.
$f (x) \to f (x) + k$		$g (x) = lo g_{2} (x) - 4,$ shifts the graph of $f (x) = lo g_{2} (x)$ down by $4$ units.
$f (x) \to f (- x) f (x) \to - f (x)$	Reflection Flips the graph over $x - axis$ if using $- x$ as argument or over $y - axis$ if using $- f (x)$	$g (x) = 4^{- x},$ reflect the graph of $f (x) = 4^{x}$ in the $y - axis$
$f (x) \to f (- x) f (x) \to - f (x)$		$g (x) = - 4^{x},$ reflect the graph of $f (x) = 4^{x}$ in the $x - axis$
$f (x) \to a f (x)$	Vertical Shrink or Stretch Graph stretches away from the $x - axis$ if $∣ a ∣ > 1$ or shrinks toward the $x - axis$ if $1 < ∣ a ∣ < 0$	$g (x) = 4 lo g_{2} (x),$ stretch the graph of $f (x) = lo g_{2} (x)$ by a factor of $4 .$
$f (x) \to a f (x)$		$g (x) = \frac{1}{2} lo g_{2} (x),$ shrinks the graph of $f (x) = lo g_{2} (x)$ by a factor of $\frac{1}{2} .$

Note, however, that a logarithmic function is only defined when its argument takes values within the interval $(0, + \infty) .$ This will remain being the case after applying the transformations.

Exercise