6.4 Graphs of Logarithmic Functions

What is the domain of $f \left(x\right) = \log_{2} \left(x + 3\right) ?$

What is the domain of $f \left(x\right) = \log_{5} \left(x - 2\right) + 1 ?$

What is the domain of $f \left(x\right) = \log \left(5 - 2 x\right) ?$

What is the domain of $f \left(x\right) = \log \left(x - 5\right) + 2 ?$

Solve $4 \ln \left(x\right) + 1 = - 2 \ln \left(x - 1\right)$ graphically. Round to the nearest thousandth.

Solve $5 \log \left(x + 2\right) = 4 - \log \left(x\right)$ graphically. Round to the nearest thousandth.

What is the vertical asymptote of $f \left(x\right) = −2 \log_{3} \left(x + 4\right) + 5 ?$

What is the vertical asymptote of $f \left(x\right) = 3 + \ln \left(x - 1\right) ?$

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

State the domain and range of the function.

$h \left(x\right) = \ln \left(\frac{1}{2} - x\right)$

State the domain and range of the function.

$h \left(x\right) = \ln \left(4 x + 17\right) - 5$

State the domain and the vertical asymptote of the function.

$f \left(x\right) = \log_{b} \left(x - 5\right)$

State the domain and the vertical asymptote of the function.

$f \left(x\right) = \log \left(3 x + 1\right)$

State the domain and the vertical asymptote of the function.

$g \left(x\right) = - \ln \left(3 x + 9\right) - 7$

State the domain, vertical asymptote, and end behavior of the function.

$f \left(x\right) = \log \left(x - \frac{3}{7}\right)$

State the domain, vertical asymptote, and end behavior of the function.

$g \left(x\right) = \ln \left(2 x + 6\right) - 5$

State the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

$h \left(x\right) = \log_{4} \left(x - 1\right) + 1$

State the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

$g \left(x\right) = \ln \left(- x\right) - 2$

State the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

$h \left(x\right) = 3 \ln \left(x\right) - 9$

$f \left(x\right) = \ln \left(x\right)$

$h \left(x\right) = \log_{5} \left(x\right)$

$f \left(x\right) = \log_{\frac{1}{3}} \left(x\right)$

$h \left(x\right) = \log_{\frac{3}{4}} \left(x\right)$

$g \left(x\right) = - \log_{4} \left(x + 2\right)$

Use a graphing calculator to find approximate solutions to each equation.

$\log \left(x - 1\right) + 2 = \ln \left(x - 1\right) + 2$

Use a graphing calculator to find approximate solutions to each equation.

$\ln \left(x - 2\right) = - \ln \left(x + 1\right)$

Use a graphing calculator to find approximate solutions to each equation.

$\frac{1}{3} \log \left(1 - x\right) = \log \left(x + 1\right) + \frac{1}{3}$

Explore and discuss the graphs of $f \left(x\right) = \log_{\frac{1}{2}} \left(x\right)$ and $g \left(x\right) = - \log_{2} \left(x\right) .$ Make a conjecture based on the result.