# 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.

What is a independent variable
a variable that does not depend on another.
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solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
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how to prove 1-sinx/cos x= cos x/-1+sin x?
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?
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?
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
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Salim
what is function.
what is polynomial
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an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
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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
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what is hyperbolic function
find volume of solid about y axis and y=x^3, x=0,y=1
3 pi/5
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what is the power rule
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.
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how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
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5x-5
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what is domain
difference btwn domain co- domain and range
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x
Verna
The set of inputs of a function. x goes in the function, y comes out.
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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
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)
pls solve it i Want to see the answer
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ok
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differentiate each term
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why do we need to study functions?
to understand how to model one variable as a direct relationship to another variable
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