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

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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.

why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
what is implicit of y³+x²y³=5 at (2,1)
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg