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Solve ln ( x 3 ) 4 ln ( x ) = 1 .

x = 1 e

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When evaluating a logarithmic function with a calculator, you may have noticed that the only options are 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 1 , b 1 .

  1. a x = b x log b a for any real number x .
    If b = e , this equation reduces to a x = e x log e a = e x ln a .
  2. log a x = log b x log b a for any real number x > 0 .
    If b = e , this equation reduces to log a x = ln x ln 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 1 , log b ( a x ) = x log b a . Therefore,

b log b ( a x ) = b x log b a .

In addition, we know that b x and log b ( x ) are inverse functions. Therefore,

b log b ( a x ) = a x .

Combining these last two equalities, we conclude that a x = b x log b a .

To prove the second property, we show that

( log b a ) · ( log a x ) = log b x .

Let u = log b a , v = log a x , and w = 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 u v = ( b u ) v = a v = x = b w .

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

Changing bases

Use a calculating utility to evaluate log 3 7 with the change-of-base formula presented earlier.

Use the second equation with a = 3 and e = 3 :

log 3 7 = ln 7 ln 3 1.77124 .

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Use the change-of-base formula and a calculating utility to evaluate log 4 6 .

1.29248

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Chapter opener: the richter scale for earthquakes

A photograph of an earthquake fault.
(credit: modification of work by Robb Hannawacker, NPS)

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 = log 10 ( A 1 A 2 ) .

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

8 7 = log 10 ( A 1 A 2 ) .

Therefore,

log 10 ( A 1 A 2 ) = 1 ,

which implies A 1 / 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

log 10 ( A 1 A 2 ) = 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 = log 10 ( A 1 A 2 ) .

Therefore, A 1 / A 2 = 10 1.7 , and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.

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Questions & Answers

What is a independent variable
Sifiso Reply
a variable that does not depend on another.
Andrew
solve number one step by step
bil Reply
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
Rochel Reply
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
Rochel Reply
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
not a bad question
Salim
what is function.
Nawaz Reply
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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