The derivative of a logarithmic function

Exponential and logarithmic functions, grade 12 (5 units) Lesson 13 of 23Learning

The derivative of a logarithmic function

The derivative of

\displaystyle \ln x

from the inverse function

$\displaystyle \ln x$ is the inverse function of $\displaystyle e^x$ , so its graph is the reflection of the graph of $\displaystyle e^x$ in the line $\displaystyle {y}={x}$ . The tangents are reflected too:

The point $\displaystyle A(1,e)$ is on the graph of $\displaystyle e^x$ , and the slope there is $\displaystyle {e^1}={e}$ .
Reflected, it goes to the point $\displaystyle B(e,1)$ on the graph of $\displaystyle \ln x$ , and the tangent $\displaystyle {y}={ex}$ goes to the tangent $\displaystyle {y}={\frac{x}{e}}$ .
The reflection swaps the change in $\displaystyle x$ and the change in $\displaystyle y$ , so the slope $\displaystyle m$ becomes $\displaystyle \frac{1}{m}$ : the slope at $\displaystyle B$ is $\displaystyle \frac{1}{e}$ .

\displaystyle f(x)=e^{x}

\displaystyle g(x)=\ln x

\displaystyle y=x

\displaystyle y=ex

\displaystyle y=\frac{x}{e}

\displaystyle A=(1,2.718)

\displaystyle B=(2.718,1)

So at every point: at the point $\displaystyle (t,e^t)$ on the graph of $\displaystyle e^x$ the slope is $\displaystyle e^t$ , and at the point $\displaystyle (e^t,t)$ on the graph of $\displaystyle \ln x$ the slope is $\displaystyle \frac{1}{e^t}$ . At that point $\displaystyle {x}={e^t}$ , so the slope is $\displaystyle \frac{1}{x}$ :

\displaystyle (\ln x)'

\displaystyle =

\displaystyle \frac{1}{x}

The derivative is defined in the domain of $\displaystyle \ln x$ , that is where $\displaystyle x>0$ .
$\displaystyle \frac{1}{x}>0$ : the function $\displaystyle \ln x$ increases throughout its domain.
$\displaystyle {(\ln x)''}={-\frac{1}{x^2}<0}$ : the graph is concave down throughout the domain.

The derivative of

\displaystyle \ln f(x)

$\displaystyle \ln f(x)$ is a composite function: the outer derivative $\displaystyle \frac{1}{f(x)}$ times the inner derivative $\displaystyle f'(x)$ .

\displaystyle \left[\ln(f(x))\right]'

\displaystyle =

\displaystyle \frac{f'(x)}{f(x)}

$\displaystyle {\left(\ln(x^2+1)\right)'}={\frac{2x}{x^2+1}}$ .
$\displaystyle {\left(\ln(3x-2)\right)'}={\frac{3}{3x-2}}$ , in the domain $\displaystyle x>\frac{2}{3}$ .
$\displaystyle {\left(\ln(5x)\right)'}={\frac{5}{5x}}={\frac{1}{x}}$ : by the laws too, $\displaystyle {\ln(5x)}={\ln5+\ln x}$ , and the derivative of the constant $\displaystyle \ln5$ is 0.

The derivative of

\displaystyle \log_a x

By the change of base, $\displaystyle {\log_a x}={\frac{\ln x}{\ln a}}$ , and $\displaystyle \ln a$ is a constant:

\displaystyle (\log_{a}x)'

\displaystyle =

\displaystyle \frac{1}{x\ln a}

$\displaystyle {(\log_2 x)'}={\frac{1}{x\ln2}}$ .
$\displaystyle {(\log x)'}={\frac{1}{x\ln10}}$ .

The derivative of

\displaystyle \ln(-x)

$\displaystyle \ln(-x)$ is defined where $\displaystyle -x>0$ , that is where $\displaystyle x<0$ . Its graph is the reflection of the graph of $\displaystyle \ln x$ in the y-axis.

By the derivative of $\displaystyle \ln f(x)$ : $\displaystyle {\left(\ln(-x)\right)'}={\frac{-1}{-x}}={\frac{1}{x}}$ .
$\displaystyle \ln x$ and $\displaystyle \ln(-x)$ have the same derivative, $\displaystyle \frac{1}{x}$ , each in its own domain.
Where $\displaystyle x<0$ , $\displaystyle \frac{1}{x}<0$ : $\displaystyle \ln(-x)$ decreases.

\displaystyle f(x)=\ln x

\displaystyle g(x)=\ln(-x)

Simplifying by the laws of logarithms before differentiating

Sometimes it pays to break up a logarithm by the laws of logarithms before differentiating. The example below differentiates $\displaystyle {f(x)}={\ln\frac{x^2}{x+1}}$ , where $\displaystyle x>0$ :

$\displaystyle {f(x)}={\ln x^2-\ln(x+1)}={2\ln x-\ln(x+1)}$ .
$\displaystyle {f'(x)}={\frac{2}{x}-\frac{1}{x+1}}$ .
Over a common denominator: $\displaystyle {f'(x)}={\frac{2(x+1)-x}{x(x+1)}}={\frac{x+2}{x(x+1)}}$ .

A tangent to the graph of a logarithm

The example below finds the equation of the tangent to the graph of $\displaystyle {f(x)}={\ln(2x-1)}$ at the point where $\displaystyle {x}={1}$ :

The point of contact: $\displaystyle {f(1)}={\ln1}={0}$ , that is $\displaystyle (1,0)$ .
The slope: $\displaystyle {f'(x)}={\frac{2}{2x-1}}$ , so $\displaystyle {f'(1)}={2}$ .
The tangent: $\displaystyle {y-0}={2(x-1)}$ , that is $\displaystyle {y}={2x-2}$ .

Exercise 1

Differentiate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The function $\displaystyle {f(x)}={\ln x+\frac{1}{x}}$ is given, in the domain $\displaystyle x>0$ .

Find $\displaystyle f'(x)$ , and write it as one fraction.

$\displaystyle f(x)=\ln x+\frac{1}{x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Find the second derivative $\displaystyle f''(x)$ , and write it as one fraction.

$\displaystyle f(x)=\ln x+\frac{1}{x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find where the graph is concave up: solve the inequality $\displaystyle f''(x)>0$ .

Mark the extremum point and the inflection point on the graph.

\displaystyle f(x)=\ln x+\frac{1}{x}

\displaystyle x=0

Exercise 3

The function $\displaystyle {f(x)}={\ln(4-x^2)}$ is given.

Find the domain of the function: solve the inequality.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\ln(4-x^{2})$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark on the graph the extremum point and the points where it meets the x-axis.

\displaystyle f(x)=\ln(4-x^{2})

\displaystyle x=-2

\displaystyle x=2

Exercise 4

A tangent to the graph of $\displaystyle {f(x)}={\ln x}$ at the point $\displaystyle (a,\ln a)$ passes through the origin.
The equation of the tangent is $\displaystyle {y-\ln a}={\frac{1}{a}(x-a)}$ .

Putting the point $\displaystyle (0,0)$ into the equation of the tangent gives $\displaystyle {-\ln a}={-1}$ , that is $\displaystyle {\ln a}={1}$ .
Solve the equation (written with $\displaystyle x$ in place of $\displaystyle a$ ).

Write the equation of the tangent.

\displaystyle f(x)=\ln x

Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$