Finite Math Examples

$\log_{b} (x) = \frac{2}{3} \cdot (\log_{b} (8 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (10)$

Step 1

Find where the expression $\frac{2}{3} \cdot (\log_{b} (8 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (10)$ is undefined.

$b \leq 0, b = 1$

Step 2

Since $\frac{2}{3} \cdot (\log_{b} (8 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (10)$ $\to$ $\infty$ as $b$ $\to$ $1$ from the left and $\frac{2}{3} \cdot (\log_{b} (8 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (10)$ $\to$ $\infty$ as $b$ $\to$ $1$ from the right, then $b = 1$ is a vertical asymptote.

$b = 1$

Step 3

$\log_{b} (x) = \frac{2}{3} \cdot (\log_{b} (8 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (10)$ is an equation of a line, which means there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 4

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 5

This is the set of all asymptotes.

Vertical Asymptotes: $b = 1$

No Horizontal Asymptotes

Step 6