6.4 Graphs of logarithmic functions  (Page 8/8)

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 range of a logarithmic function?

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

Consider the general logarithmic function $f(x)={\log }_b\left(x\right).$ Why can’t $x$ be zero?

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

$f(x)={\log }_3\left(x+4\right)$

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

$g(x)={\log }_5\left(2x+9\right)-2$

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

$f(x)={\log }_2\left(12-3x\right)-3$

$f(x)={\log }_b(x-5)$

$g(x)=\ln (3-x)$

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

$f(x)=3\log (-x)+2$

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

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

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

$h(x)=-\log \left(3x-4\right)+3$

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

$f(x)={\log }_3\left(15-5x\right)+6$

$h(x)={\log }_4\left(x-1\right)+1$

$f(x)=\log \left(5x+10\right)+3$

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

$f(x)={\log }_2\left(x+2\right)-5$

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

$d(x)=\log \left(x\right)$

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

$g(x)={\log }_2\left(x\right)$

$h(x)={\log }_5\left(x\right)$

$j(x)={\log }_{25}\left(x\right)$

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

$g(x)={\log }_2\left(x\right)$

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

$f(x)=\log (x)$ and $g(x)={10}^x$

$f(x)=\log (x)$ and $g(x)={\log }_{\frac{1}{2}}(x)$

$f(x)={\log }_4(x)$ and $g(x)=\ln (x)$

$f(x)=e^x$ and $g(x)=\ln (x)$

$f(x)={\log }_4\left(-x+2\right)$

$g(x)=-{\log }_4\left(x+2\right)$

$h(x)={\log }_4\left(x+2\right)$

$f(x)={\log }_2(x+2)$

$f(x)=2\log (x)$

$f(x)=\ln (-x)$

$g(x)=\log \left(4x+16\right)+4$

$g(x)=\log \left(6-3x\right)+1$

$h(x)=-\frac{1}{2}\ln \left(x+1\right)-3$

Use $y={\log }_2(x)$ as the parent function.

Use $f(x)={\log }_3(x)$ as the parent function.

Use $f(x)={\log }_4(x)$ as the parent function.

Use $f(x)={\log }_5(x)$ as the parent function.

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

$\log \left(2x-3\right)+2=-\log \left(2x-3\right)+5$

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

$2\ln \left(5x+1\right)=\frac{1}{2}\ln \left(-5x\right)+1$

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

Let $b$ be any positive real number such that $b\ne 1.$ What must ${\log }_{b}1$ be equal to? Verify the result.

Explore and discuss the graphs of $f(x)={\log }_{\frac{1}{2}}\left(x\right)$ and $g(x)=-{\log }_2\left(x\right).$ Make a conjecture based on the result.

Prove the conjecture made in the previous exercise.

What is the domain of the function $f(x)=\ln \left(\frac{x+2}{x-4}\right)?$ Discuss the result.

Use properties of exponents to find the x -intercepts of the function $f(x)=\log \left(x^2+4x+4\right)$ algebraically. Show the steps for solving, and then verify the result by graphing the function.