# 1.5 Exponential and logarithmic functions  (Page 5/17)

 Page 5 / 17

Solve $\text{ln}\left({x}^{3}\right)-4\phantom{\rule{0.1em}{0ex}}\text{ln}\left(x\right)=1.$

$x=\frac{1}{e}$

When evaluating a logarithmic function with a calculator, you may have noticed that the only options are ${\text{log}}_{10}$ or log, called the common logarithm , or ln , which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base $b.$ If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.

## Rule: change-of-base formulas

Let $a>0,b>0,$ and $a\ne 1,b\ne 1.$

1. ${a}^{x}={b}^{x{\text{log}}_{b}a}$ for any real number $x.$
If $b=e,$ this equation reduces to ${a}^{x}={e}^{x{\text{log}}_{e}a}={e}^{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}a}.$
2. ${\text{log}}_{a}x=\frac{{\text{log}}_{b}x}{{\text{log}}_{b}a}$ for any real number $x>0.$
If $b=e,$ this equation reduces to ${\text{log}}_{a}x=\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}x}{\text{ln}\phantom{\rule{0.1em}{0ex}}a}.$

## Proof

For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base $b>0,b\ne 1,{\text{log}}_{b}\left({a}^{x}\right)=x{\text{log}}_{b}a.$ Therefore,

${b}^{{\text{log}}_{b}\left({a}^{x}\right)}={b}^{x{\text{log}}_{b}a}.$

In addition, we know that ${b}^{x}$ and ${\text{log}}_{b}\left(x\right)$ are inverse functions. Therefore,

${b}^{{\text{log}}_{b}\left({a}^{x}\right)}={a}^{x}.$

Combining these last two equalities, we conclude that ${a}^{x}={b}^{x{\text{log}}_{b}a}.$

To prove the second property, we show that

$\left({\text{log}}_{b}a\right)·\left({\text{log}}_{a}x\right)={\text{log}}_{b}x.$

Let $u={\text{log}}_{b}a,v={\text{log}}_{a}x,$ and $w={\text{log}}_{b}x.$ We will show that $u·v=w.$ By the definition of logarithmic functions, we know that ${b}^{u}=a,{a}^{v}=x,$ and ${b}^{w}=x.$ From the previous equations, we see that

${b}^{uv}={\left({b}^{u}\right)}^{v}={a}^{v}=x={b}^{w}.$

Therefore, ${b}^{uv}={b}^{w}.$ Since exponential functions are one-to-one, we can conclude that $u·v=w.$

## Changing bases

Use a calculating utility to evaluate ${\text{log}}_{3}7$ with the change-of-base formula presented earlier.

Use the second equation with $a=3$ and $e=3\text{:}$

${\text{log}}_{3}7=\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}7}{\text{ln}\phantom{\rule{0.1em}{0ex}}3}\approx 1.77124.$

Use the change-of-base formula and a calculating utility to evaluate ${\text{log}}_{4}6.$

$1.29248$

## Chapter opener: the richter scale for earthquakes

In 1935, Charles Richter developed a scale (now known as the Richter scale ) to measure the magnitude of an earthquake . The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude ${R}_{1}$ on the Richter scale and a second earthquake with magnitude ${R}_{2}$ on the Richter scale. Suppose ${R}_{1}>{R}_{2},$ which means the earthquake of magnitude ${R}_{1}$ is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If ${A}_{1}$ is the amplitude measured for the first earthquake and ${A}_{2}$ is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:

${R}_{1}-{R}_{2}={\text{log}}_{10}\left(\frac{{A}_{1}}{{A}_{2}}\right).$

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,

$8-7={\text{log}}_{10}\left(\frac{{A}_{1}}{{A}_{2}}\right).$

Therefore,

${\text{log}}_{10}\left(\frac{{A}_{1}}{{A}_{2}}\right)=1,$

which implies ${A}_{1}\text{/}{A}_{2}=10$ or ${A}_{1}=10{A}_{2}.$ Since ${A}_{1}$ is 10 times the size of ${A}_{2},$ we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation

${\text{log}}_{10}\left(\frac{{A}_{1}}{{A}_{2}}\right)=8-6=2.$

Therefore, ${A}_{1}=100{A}_{2}.$ That is, the first earthquake is 100 times more intense than the second earthquake.

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:

$9-7.3={\text{log}}_{10}\left(\frac{{A}_{1}}{{A}_{2}}\right).$

Therefore, ${A}_{1}\text{/}{A}_{2}={10}^{1.7},$ and we conclude that the earthquake in Japan was approximately $50$ times more intense than the earthquake in Haiti.

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
do u mean fundamental theorem ?
Roha
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