Logarithms

Hong Kong

Stage 4 - Stage 5

A major communications company found that the more they spend on advertising, the higher their revenue. Their sales revenue, in thousands of dollars, is given by $R=10+20\log_4\left(x+1\right)$`R`=10+20`l``o``g`4(`x`+1), where $x$`x` represents the amount they spend on advertising (in thousands of dollars).

a

Determine their sales revenue if they spend no money on advertising.

b

Determine their sales revenue if they spend $\$14000$$14000 on advertising.

Give your answer to the nearest thousand $dollars$`d``o``l``l``a``r``s`.

c

Would you say that every extra $\$1000$$1000 spent on advertising becomes more or less effective in terms of raising revenue?

more effective: sales revenue increases as advertising spending increases.

A

less effective: every extra $\$1000$$1000 spent on advertising raises the sales revenue by less and less

B

Easy

Approx 4 minutes

Researchers conducted a test to determine how well information is retained through the method of rote learning. To do this, they asked students to memorise mathematical formulae in the lead up to the first test, and then study no further. They continued to test them once a month over $7$7 months. They found that the average student’s test scores ($P$`P`) decreased over time ($t$`t` months), but at a slowing rate.

The number of registered nurses working in hospitals $t$`t` years after the year $2002$2002 can be modelled by the equation $N=28\log_4\left(t+2\right)$`N`=28`l``o``g`4(`t`+2), where $N$`N` represents the number of nurses in thousands.

Graph $y=\log_4x$`y`=`l``o``g`4`x`, $y=\log_{25}x$`y`=`l``o``g`25`x` and $y=\log_{100}x$`y`=`l``o``g`100`x` in the same viewing window using a graphing calculator. Use this graph to answer the following questions.

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