# How do you graph y=log_5(x-1)+3?

Nov 22, 2017

See below.

#### Explanation:

$y = {\log}_{5} \left(x - 1\right) + 3$

Remember: ${\log}_{a} x = \ln \frac{x}{\ln} a$

$\therefore {\log}_{5} \left(x - 1\right) = \ln \frac{x - 1}{\ln} 5$

Hence, ${\log}_{5} \left(x - 1\right)$ can be graphed as the function transformed to natural logs above. As shown below.

graph{ln(x-1)/ln5 [-14.24, 14.23, -7.12, 7.12]}

The constant term $+ 3$ simply shifts the graph 3 units positive ("up") the $y -$axis.

The resultant graph of $y$ is shown below.

graph{ln(x-1)/ln5 +3 [-14.24, 14.23, -7.12, 7.12]}