The Role of the Parameters in a Logarithmic Function

Adding the parameters |a,| |b,| |h,| and |k| to the basic form |f(x)= \log_c x| results in what is called the standard form (also called the transformed form) of the logarithmic function.

In the following interactive animation, experiment with the values of parameters |a|, |b|, |c|, |h,| and |k| of the logarithmic function. Observe the modifications which take place on the transformed curve (in black) compared to the base function with |c=2| (in green). Observe the effect of modifying the parameters on the function’s properties.

When |a>1|

The graph stretches vertically compared to the base function. The greater the absolute value of the parameter |a|, the farther away the curve of the log function is from the |x|-axis.

When |0< a <1|

The graph contracts vertically relative to the base function. The smaller the absolute value of the parameter |a| (closer to |0|), the closer the curve of the log function is to the |x|-axis.

When |a| is positive |a>0|

The curve of the logarithmic function is increasing.

When |a| is negative |a<0|

The curve of the logarithmic function is decreasing.

When |b>1|

The graph contracts horizontally relative to the base function. The larger the absolute value of the parameter |b|, the more the curve of the logarithm function approaches the |y|-axis.

When |0<b<1|

The graph stretches horizontally relative to the base function. The smaller (close to |0|) the absolute value of the parameter |b|, the farther the curve of the logarithm function is from the |y|-axis.

When |b| is positive |b>0|

The curve of the logarithmic function is entirely located to the right of the asymptote.

When |b| is negative |b<0|

The curve of the logarithmic function is entirely located to the left of the asymptote.

The value of |c| represents the base of the function, that is the multiplying factor present in the exponential function.

When |c>1|

The basic function is increasing.

When |0<c<1|

The basic function is decreasing.

When |h| is positive |h>0|

The curve of the logarithmic function moves to the right.

When |h| is negative |h<0|

The curve of the logarithmic function moves to the left.

When |k| is positive |k>0|

The curve of the logarithmic function moves upwards.

When |k| is negative |k<0|

The curve of the logarithmic function moves downward.

If |c>1|

	|a>0|	|a<0|
|b>0|
|b<0|

If |0<c<1|

	|a>0|	|a<0|
|b>0|
|b<0|

As seen in the summary table above, certain combinations of parameters give the same result. For example, having |c>1,| |a>0,| and |b>0| is equivalent to having |0<c<1,| |a<0| et |b>0.| For this reason, the equation of a logarithmic function in standard form is often simplified by omitting the parameters |a| and |k.| ||\begin{align} f(x) &= a\log_c b(x-h)+k \\ \Rightarrow \ f(x) &= \log_c b(x-h) \end{align}|| For the simplified equation of a logarithmic function, the summary table would be the following.

	|b>0|	|b<0|
|c>1|
|0<c<1|