As we have seen with other functions, we can use our knowledge of differentiating logarithmic functions to find the equation of tangents and normals to such functions.

Differentiating $y=\ln f\left(x\right)$y=lnf(x)

$\frac{d}{dx}\ln f\left(x\right)$ddxlnf(x)

$=$=

$\frac{f'\left(x\right)}{f\left(x\right)}$f′(x)f(x)

When finding equations of tangents and normals, remember that the gradient of the tangent at any point on a function is given by the first derivative of the function. The following formulas are also useful:

Formulas for tangents and normals

Given the gradient $m$m and a point $\left(x_1,y_1\right)$(x1,y1), the equation of a line can be found using the point gradient formula:

$y-y_1=m\left(x-x_1\right)$y−y1=m(x−x1)

Given the gradient of the tangent $m_1$m1 at a point, the gradient of the normal $m_2$m2 is:

$m_2=-\frac{1}{m_1}$m2=−1m1

Worked example

Example 1

Find the gradient of the tangent to the curve $y=\ln\left(4x-8\right)$y=ln(4x−8) at the point where $x=3$x=3.

Solution: Finding the derivative of $y=\ln\left(4x-8\right)$y=ln(4x−8) gives:

$\frac{dy}{dx}$`dydx`	$=$=	$\frac{4}{4x-8}$44`x`−8
	$=$=	$\frac{1}{x-2}$1`x`−2

So at $x=3$x=3, the gradient of the tangent is:

$\frac{dy}{dx}$`dydx`	$=$=	$\frac{1}{3-2}$13−2
	$=$=	$1$1

Practice question

Question 1

Consider the function $f\left(x\right)=\ln x$f(x)=lnx.

Determine the $x$x value of the $x$x-intercept.
Determine the equation of the tangent line to the curve at the point where it crosses the $x$x-axis.
Determine the equation of the normal line to the curve at the point where it crosses the $x$x-axis.

Determining the characteristics of, and sketching logarithmic functions

The first and second derivatives of logarithmic functions can be used to determine concavity, and regions where a logarithmic function is increasing or decreasing. Additionally, any stationary points and points of inflection can be found. Using these tools we can sketch functions involving logarithms.

Practice questions

Question 2

Consider the graph of $y=\ln x$y=lnx.

Loading Graph...

Is the function increasing or decreasing?
Increasing
A
Decreasing
B
Is the gradient to the curve negative at any point on the curve?
No
A
Yes
B
Which of the following best completes this sentence?

"As $x$x increases, the gradient of the tangent..."
decreases at a constant rate.
A
increases at an increasing rate.
B
increases at a constant rate.
C
decreases at an increasing rate.
D
increases at a decreasing rate.
E
decreases at a decreasing rate.
F
Which of the following best completes the sentence?

"As $x$x gets closer and closer to $0$0, the gradient of the tangent..."
increases towards a fixed value.
A
decreases towards $-\infty$−∞.
B
decreases towards $0$0.
C
increases towards $\infty$∞.
D
We have found that the gradient function must be a strictly positive function, and it must also be a function that decreases at a decreasing rate. What kind of function could it be?
Quadratic, of the form $y'=ax^2$y′=ax2.
A
Exponential, of the form $y'=a^{-x}$y′=a−x.
B
Linear, of the form $y=ax$y=ax.
C
Hyperbolic, of the form $y'=\frac{a}{x}$y′=ax.
D

Question 3

Consider the function $f\left(x\right)=x-\ln x$f(x)=x−lnx.

Find the values of $x$x where $f'\left(x\right)=0$f′(x)=0.
Find $f''\left(x\right)$f′′(x).
Determine the nature of the stationary point.
Minimum
A
Point of inflection
B
Maximum
C
Which of the following best describes the behaviour of this function?
As $x$x$\to$→$\infty$∞, $y$y$\to$→$-\infty$−∞.

As $x$x$\to$→$0$0, $y$y$\to$→$-\infty$−∞.
A
As $x$x$\to$→$\infty$∞, $y$y$\to$→$0$0.

As $x$x$\to$→$0$0, $y$y$\to$→$\infty$∞.
B
As $x$x$\to$→$\infty$∞, $y$y$\to$→$\infty$∞.

As $x$x$\to$→$0$0, $y$y$\to$→$0$0.
C
As $x$x$\to$→$\infty$∞, $y$y$\to$→$\infty$∞.

As $x$x$\to$→$0$0, $y$y$\to$→$\infty$∞.
D
Which is the correct sketch of the graph?
Loading Graph...
A
Loading Graph...
B
Loading Graph...
C
Loading Graph...
D

Outcomes

0606C14.3B

Use the derivatives of the standard functions sinx, cos x, tan x, together with constant multiples, sums and composite functions.

0606C14.3C

Use the derivatives of the standard functions e^x, ln x, together with constant multiples, sums and composite functions.

11.16 Applications of the differentiation of logarithmic functions

Tangents and normals to logarithmic functions