Logarithms

Hong Kong

Stage 4 - Stage 5

Consider the functions:

$f\left(x\right)=\log_2x$`f`(`x`)=`l``o``g`2`x`, $g\left(x\right)=\log_2\left(x+6\right)$`g`(`x`)=`l``o``g`2(`x`+6), $h\left(x\right)=\log_2x+6$`h`(`x`)=`l``o``g`2`x`+6 and $p\left(x\right)=6\log_2x$`p`(`x`)=6`l``o``g`2`x`

a

Determine each of the function values for when $x=2$`x`=2.

$f$f$\left(2\right)$(2) |
$=$= | $\editable{}$ |

$g$g$\left(2\right)$(2) |
$=$= | $\editable{}$ |

$h$h$\left(2\right)$(2) |
$=$= | $\editable{}$ |

$p$p$\left(2\right)$(2) |
$=$= | $\editable{}$ |

b

Which of the following is true for the relationship between the function $f\left(x\right)=\log_2x$`f`(`x`)=`l``o``g`2`x` and the transformation $g\left(x\right)=\log_2\left(x+6\right)$`g`(`x`)=`l``o``g`2(`x`+6)?

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $g\left(x\right)$`g`(`x`) becomes very large.

A

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $g\left(x\right)$`g`(`x`) stays the same.

B

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $g\left(x\right)$`g`(`x`) becomes very small.

C

c

Which of the following is true for the relationship between the function $f\left(x\right)=\log_2x$`f`(`x`)=`l``o``g`2`x` and the transformation $h\left(x\right)=\log_2x+6$`h`(`x`)=`l``o``g`2`x`+6?

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $h\left(x\right)$`h`(`x`) becomes very large.

A

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $h\left(x\right)$`h`(`x`) stays the same.

B

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $h\left(x\right)$`h`(`x`) becomes very small.

C

d

Which of the following is true for the relationship between the function $f\left(x\right)=\log_2x$`f`(`x`)=`l``o``g`2`x` and the transformation $p\left(x\right)=6\log_2x$`p`(`x`)=6`l``o``g`2`x`?

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $p\left(x\right)$`p`(`x`) stays the same.

A

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $p\left(x\right)$`p`(`x`) becomes very small.

B

As $x$`x` becomes very large, the difference between the function values of $f\left(x\right)$`f`(`x`) and $p\left(x\right)$`p`(`x`) becomes very large.

C

Easy

Approx 3 minutes

The graph of $y=\log_4x$`y`=`l``o``g`4`x` has a vertical asymptote at $x=0$`x`=0. By considering the transformations that have taken place, state the equation of the asymptote of:

Write a fully simplified equation for when the graph of $y=\log_4x$`y`=`l``o``g`4`x` is translated seven units downward, six units to the left, and then reflected across the $x$`x`-axis.

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