Interactive Logarithmic Functions HTML5 Applet

Interactive Applet: Graph $f(x)=a\log_B(b(x+c))+d$

Tutorial — Part 1: Basics of Logarithmic Graphs

1. Set $a = 1$, $b = 1$, $c = 0$, $d = 0$ and $B = 2$. Check a few points on the graph such as $\log_2 1 = 0$, $\log_2 2 = 1$, $\log_2 4 = 2$. Use zoom in and out if necessary.
2. Keep the same values for $a$, $b$, $c$ and $d$ as above and set $B = 4$. Check points such as $\log_4 1 = 0$, $\log_4 4 = 1$, $\log_4 16 = 2$.
3. Keep the same values for $a$, $b$, $c$ and $d$ as above and set $B = 0.5$. Check points such as $\log_{0.5} 1 = 0$, $\log_{0.5} 2 = -1$, $\log_{0.5} 4 = -2$, $\log_{0.5} 8 = -3$.

Tutorial — Part 2: Transformations & Analysis

1. Investigate the base $B$. Set $a=1$, $b=1$, $c=0$, $d=0$. Try bases between 0 and 1 and greater than 1, then describe how the graph changes.
   View Solution

   For $f(x) = \log_B(x)$:

   • If $0 < B < 1$, the function is decreasing.
   • If $B > 1$, the function is increasing.

   This follows because the logarithm is the inverse of the exponential function, whose monotonicity depends directly on its base.
2. Effect of parameter $a$ (vertical scaling). Set $B = e$, $b = 1$, $c = 0$, $d = 0$.
   View Solution

   The function is $f(x) = a \log_B(x)$. As $a$ increases, the graph is stretched vertically; as $a$ decreases (but remains nonzero), it is compressed vertically. If $a < 0$, the graph is also reflected across the $x$-axis.
3. Effect of parameter $b$ (horizontal scaling). Set $a = 1$, $c = 0$, $d = 0$, $B = e$.
   View Solution

   The function becomes $f(x) = \log_B(bx)$. As $b$ increases, the graph is compressed horizontally; as $b$ decreases (but remains positive), it is stretched horizontally.
4. Effect of parameters $c$ (horizontal shift) and $d$ (vertical translation). Set $B=e$, $a=1$, $b=1$.
   View Solution

   The function is $f(x) = \log_B(x + c) + d$.

   • If $c > 0$, the graph shifts left; if $c < 0$, it shifts right.
   • If $d > 0$, the graph shifts up; if $d < 0$, it shifts down.
5. How do parameters $b$ and $c$ affect the domain? Explain analytically.
   View Solution

   For $f(x) = a \log_B(b(x + c)) + d$, the argument must be strictly positive: $$ b(x + c) > 0 $$

   • If $b > 0$, the domain is $x > -c$.
   • If $b < 0$, the inequality flips, so the domain is $x < -c$.
6. Which parameter(s) affect the $x$-intercept? Is there always one?
   View Solution

   The $x$-intercept satisfies $f(x) = 0$: $$ a \log_B(b(x + c)) + d = 0 $$ $$ \log_B(b(x + c)) = -\frac{d}{a} $$ $$ b(x + c) = B^{-d/a} $$ $$ x = \frac{1}{b}B^{-d/a} - c $$ An $x$-intercept exists when $a \neq 0$, $b \neq 0$, and the resulting value lies in the domain.
7. Which parameter(s) affect the $y$-intercept? Is there always one?
   View Solution

   The $y$-intercept is $f(0) = a\log_B(bc) + d$. It exists only when $bc > 0$, so that the argument at $x=0$ is strictly positive.
8. Which parameter(s) affect the vertical asymptote?
   View Solution

   The asymptote occurs where the argument equals zero: $$ b(x + c) = 0 \implies x = -c. $$ The vertical asymptote is therefore the line $x = -c$, controlled directly by the parameter $c$.

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