Informative line

Different Operations On Log Function Like Inverse Limits

Finding the Inverse of a Function & Limit of Expression involves Log Function, practice to solve inequalities containing logarithmic function and domain and range of log functions.

Domain and Range of Logarithmic Functions

• Domain of $$log_ax$$ is $$x>0$$ or $$(0, \infty)$$. [ for all valued  $$'a'$$ ]
• Range of $$log_ax$$ is R or $$(-\infty, \infty)$$ (It takes all possible value)
• $$log_ax=f(x)$$ is a one to one function for all values of $$'a'$$.
• To find the domain of $$f(x)=log_a\;(g(x))$$, when $$g(x)$$ is some expression in $$x$$ , we solve the inequality $$g(x)>0$$

Find the domain of $$f(x)=log_5(x^2-16)$$

A $$(-\infty, 2)\cup(5,\infty)$$

B $$(-\infty, 6)\cup(10,\infty)$$

C $$(-\infty, -4)\cup(4,\infty)$$

D $$(2,7)$$

×

Domain of $$f(x)=log_a\Big(g(x)\Big)$$ is $$g(x)>0$$.

$$\therefore \;$$Domain of $$log_5\Big(x^2-16\Big)$$ is $$x^2-16>0$$

$$\Rightarrow (x-4)(x+4)>0$$

Interval $$x+4$$ $$x-4$$ $$(x+4)(x-4)$$
$$x>4$$ + + +
$$-4<x<4$$ +
$$x<-4$$ +

$$\Rightarrow x\in(-\infty,-4)\cup(4,\infty)$$

$$\therefore \;$$ Domain of $$'f'$$ is $$(-\infty, -4)\cup(4,\infty)$$.

Find the domain of $$f(x)=log_5(x^2-16)$$

A

$$(-\infty, 2)\cup(5,\infty)$$

.

B

$$(-\infty, 6)\cup(10,\infty)$$

C

$$(-\infty, -4)\cup(4,\infty)$$

D

$$(2,7)$$

Option C is Correct

Finding the Inverse of a Function whose Expression involves Log Function

• To find the inverse of a function:
1. Let $$y = f(x)$$ be the given function.
2. Solve $$x$$ in term of $$y$$.
3. Interchange $$x$$ and $$y$$, the new $$y$$ obtained is the required inverse.

Find the inverse function of the function  $$f(x)=log_5(6+log_5x)$$.

A $$f^{-1}(x)=5^{(5^{x}-6)}$$

B $$f^{-1}(x)=6^{(5^{x}-6)}$$

C $$f^{-1}(x)=7^x$$

D $$f^{-1}(x)=\ell n\,x$$

×

$$f(x)=log_5(6+log_5x)$$

Step 1 :

$$y=log_5(6+log_5x)$$

Step 2 :

$$5^y=log_5\,x+6\Rightarrow log_5\,x=5^y-6$$

$$x=5^{(5^y-6)}$$

Step 3 :

$$y=5^{(5^x-6)}=f^{-1}(x)$$

Find the inverse function of the function  $$f(x)=log_5(6+log_5x)$$.

A

$$f^{-1}(x)=5^{(5^{x}-6)}$$

.

B

$$f^{-1}(x)=6^{(5^{x}-6)}$$

C

$$f^{-1}(x)=7^x$$

D

$$f^{-1}(x)=\ell n\,x$$

Option A is Correct

Finding Limit of Expression that involves Log Function

If $$a>1$$, graph of $$\,log_ax$$ is

• Observe that,

$$\lim\limits_{x\rightarrow\infty}\,log_ax=\infty$$ ...(1)

$$\lim\limits_{x\rightarrow0^+}\,log_ax=-\infty$$ ...(2)

• This means that log of very large quantities take very large values and log of small positive quantities take very large negative values.

If $$0<a<1$$, graph of $$log_ax$$ is:

• Observe that

$$\lim\limits_{x\rightarrow\infty}\,log_ax=-\infty$$ ...(3)

$$\lim\limits_{x\rightarrow0^+}\,log_ax=\infty$$ ...(4)

• This means that log of very large positive quantities take very large negative values and log of small positive quantities take very large positive values.

Evaluate   $$\lim\limits_{x\rightarrow3^+}\,log_2\,(5x-15)$$.

A $$\dfrac {5}{3}$$

B 0

C $$-\infty$$

D $$\infty$$

×

Here, $$a=2,\;5x-15$$ will take small positive values where $$x$$take $$3^+$$i.e. values just larger than 3.

$$\therefore\;\lim\limits_{x\rightarrow3^+}\,log_2\,(5x-15)$$

$$=\lim\limits_{x\rightarrow3^+}\,log_2\,y$$ (where $$y$$ is small positive )

$$=-\infty$$ (by (2))

Evaluate   $$\lim\limits_{x\rightarrow3^+}\,log_2\,(5x-15)$$.

A

$$\dfrac {5}{3}$$

.

B

0

C

$$-\infty$$

D

$$\infty$$

Option C is Correct

More Limits on Log Function (infinity minus infinity$$\infty -\infty$$ form)

• To evaluate limits of the form $$\lim\limits_{x\rightarrow \infty}\Big(log\,f(x)-log\,g(x)\Big)$$, where $$f(x)$$ and $$g(x)$$ are both tending to $$\infty$$ ($$\infty -\infty$$ form) we first use properties of log to get $$\lim\limits_{x\rightarrow \infty}\,log\left ( \dfrac {f(x)}{g(x)}\right)$$.
• Then according to the degree of  $$f(x)$$ and $$g(x)$$ ,we give the answer.

Evaluate  $$\lim\limits_{x\rightarrow \infty}\Big(\ell n\,(x+7)-\ell n\,(x+5)\Big)$$.

A $$\dfrac {1}{2}$$

B $$-7$$

C 5

D $$0$$

×

$$\lim\limits_{x\rightarrow \infty}\Big(\ell n\,(x+7)-\ell n\,(x+5)\Big)$$

$$=\lim\limits_{x\rightarrow \infty}\dfrac {(x+7)}{(x+5)}$$      (Use property of log)

$$=\lim\limits_{x\rightarrow \infty}\dfrac {\Big(1+\dfrac {7}{x}\Big)} {\Big(1+\dfrac {5}{x}\Big)}=\ell \,n\,1=0$$    $$\left (\because\dfrac {7}{x}\,,\dfrac {5}{x} \rightarrow 0 \right)$$

Evaluate  $$\lim\limits_{x\rightarrow \infty}\Big(\ell n\,(x+7)-\ell n\,(x+5)\Big)$$.

A

$$\dfrac {1}{2}$$

.

B

$$-7$$

C

5

D

$$0$$

Option D is Correct

Solving Inequalities Containing Log Function

• If $$a > 1$$, then
1. $$log_ax>b\Rightarrow x>a^b \Rightarrow x\in(a^b, \infty)$$
2. $$log_ax<b\Rightarrow x<a^b \Rightarrow x\in(0, a^b)$$ because $$x$$ has to be positive for its log to be defined.

e.g.

1. $$log_2x>3\Rightarrow x>2^3 \text { or } x > 8$$
2. $$log_{1/2}x>3\Rightarrow x<\Big(\dfrac {1}{2}\Big)^3 \text { or } x < \dfrac {1}{8}$$ .

In (1) base 2 is greater than 1  whereas in (2) base $$\dfrac {1}{2}$$  is less than 1 .

Solve for $$x$$ the inequality $$2-5\,\ell n\,x>7$$.

A $$x\in\left (\dfrac {1}{e},\infty\right)$$

B $$x\in(1,e)$$

C $$x\in\left (0,\;\dfrac {1}{e}\right)$$

D $$x\in(e, \infty)$$

×

$$2-5\,\ell n\,x>7\;$$

$$\Rightarrow 5\,\ell n\,x<2-7$$

$$\Rightarrow \ell n\,x<\dfrac {-5}{5}\;$$

$$\Rightarrow \ell n\,x<-1$$

$$\therefore \;x<e^{-1} \;$$

$$\Rightarrow x< \dfrac {1}{e}$$

$$\Rightarrow x\in\left ( 0,\dfrac {1}{e} \right)$$      because $$x$$has to be positive.

Solve for $$x$$ the inequality $$2-5\,\ell n\,x>7$$.

A

$$x\in\left (\dfrac {1}{e},\infty\right)$$

.

B

$$x\in(1,e)$$

C

$$x\in\left (0,\;\dfrac {1}{e}\right)$$

D

$$x\in(e, \infty)$$

Option C is Correct

Graph of logax v/s x

• If  $$a>1$$ then $$log_ax$$ is an increasing function of $$x$$.

• If  $$0<a<1$$ then $$log_ax$$ is a decreasing function of $$x$$.

• The point $$(1, 0)$$ is on every $$y=log_ax$$ curve, for all allowed values of $$'a'$$.
• For  $$a>1$$, the graph of  $$log_ax$$ for different values of $$'a'$$ will be as shown in figure.

• As 'a' increases the rate of increase of $$log$$ function decreases.

• If  $$0<a<1$$ , then the graph of $$log_ax$$ for different values of $$'a'$$ will be as shown in figure.

Which of the following graphs shows the correct sequence of $$log_ax$$ graphs?

A

B

C

D

×

If a > 1,  $$log_ax$$ increases, the rate of increase decreases with increasing values of $$'a'$$.

$$\therefore$$ (A) is correct option.

Which of the following graphs shows the correct sequence of $$log_ax$$ graphs?

A
B
C
D

Option A is Correct