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6.09 Further applications of logarithms

Lesson

Using our knowledge of the natural logarithm function and its inverse relationship with the exponential function, we can apply our newly acquired tools in solving equations and calculus to a wide range of applications. We can now find algebraic solutions for exponential problems where previously we required technology, as well as being able to apply calculus to general exponential functions. 

Keep the following rules in mind as we look at some examples.

Function Derivative 
$e^{f\left(x\right)}$ef(x) $f\ '\left(x\right)e^{f\left(x\right)}$f (x)ef(x)
$\ln\left(f\left(x\right)\right)$ln(f(x)) $\frac{f\ '\left(x\right)}{f\left(x\right)}$f (x)f(x)

 

Function $f\left(x\right)$f(x) Integral $\int f\left(x\right)dx$f(x)dx
$e^{ax+b}$eax+b $\frac{1}{a}e^{ax+b}+C$1aeax+b+C
$\frac{1}{ax+b}$1ax+b $\frac{1}{a}\ln\left(ax+b\right)+C$1aln(ax+b)+C, for $ax+b>0$ax+b>0

 

Worked examples

Example 1

The gradient of a curve at any point is given by $f\ '\left(x\right)=\frac{6}{x-3}$f (x)=6x3, and the curve passes through the point $\left(4,2\right)$(4,2) and is defined for $x>3$x>3.

Find an expression for the function $f(x)$f(x).

Think: Use integration to find the family of curves with the gradient function $f\ '\left(x\right)=\frac{6}{x-3}$f (x)=6x3 and then use the given point to solve for the constant of integration, $C$C.

Do:

$f\left(x\right)$f(x) $=$= $\int\frac{6}{x-3}dx$6x3dx
  $=$= $6\int\frac{1}{x-3}dx$61x3dx
  $=$= $6\ln\left(x-3\right)+C$6ln(x3)+C
     

Using the given point $\left(4,2\right)$(4,2), substitute into the function to solve for $C$C.

$2$2 $=$= $6\ln\left(4-3\right)+C$6ln(43)+C
$2$2 $=$= $6\ln1+C$6ln1+C
$\therefore$    $C$C $=$= $2$2
     

The function is given by $f\left(x\right)=6\ln\left(x-3\right)+2$f(x)=6ln(x3)+2.

 

Practice questions

Question 1

A plane takes off from an airport at sea level and its altitude $h$h in metres, $t$t minutes after taking off, is given by $h=600\ln\left(t+1\right)$h=600ln(t+1).

  1. Exactly $t$t minutes after taking off, at what rate is the plane ascending?

  2. Hence, what is the rate of ascent at exactly $4$4 minutes after take off?

  3. How would you describe the ascent of the plane?

    Ascending, at an increasing rate.

    A

    Ascending, but at a decreasing rate.

    B

Question 2

A circus tent is $7$7 m high and has a radius of $6$6 m. The equation to describe the curved roof of the tent is $y=\frac{7}{x+1}$y=7x+1, as shown in the graph.

Calculate the cross-sectional area of the tent. Leave your answer in exact form.

 

Outcomes

3.1.1.6

identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

3.2.2.4

use logarithmic functions and their derivatives to solve practical problems

3.3.3.1

calculate the area under a curve

3.3.3.3

calculate the area between curves with and without technology

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