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log 15 ( a ) = b

15 b = a

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log 13 ( 142 ) = a

13 a = 142

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For the following exercises, rewrite each equation in logarithmic form.

n 4 = 103

log n ( 103 ) = 4

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y x = 39 100

log y ( 39 100 ) = x

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For the following exercises, solve for x by converting the logarithmic equation to exponential form.

log 2 ( x ) = 3

x = 2 3 = 1 8

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log 3 ( x ) = 3

x = 3 3 = 27

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log 9 ( x ) = 1 2

x = 9 1 2 = 3

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log 6 ( x ) = 3

x = 6 3 = 1 216

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For the following exercises, use the definition of common and natural logarithms to simplify.

e ln ( 10.125 ) + 4

14.125

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Numeric

For the following exercises, evaluate the base b logarithmic expression without using a calculator.

For the following exercises, evaluate the common logarithmic expression without using a calculator.

2 log ( 100 3 )

12

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For the following exercises, evaluate the natural logarithmic expression without using a calculator.

ln ( e 0.225 ) 3

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Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

Extensions

Is x = 0 in the domain of the function f ( x ) = log ( x ) ? If so, what is the value of the function when x = 0 ? Verify the result.

No, the function has no defined value for x = 0. To verify, suppose x = 0 is in the domain of the function f ( x ) = log ( x ) . Then there is some number n such that n = log ( 0 ) . Rewriting as an exponential equation gives: 10 n = 0 , which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function f ( x ) = log ( x ) .

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Is f ( x ) = 0 in the range of the function f ( x ) = log ( x ) ? If so, for what value of x ? Verify the result.

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Is there a number x such that ln x = 2 ? If so, what is that number? Verify the result.

Yes. Suppose there exists a real number x such that ln x = 2. Rewriting as an exponential equation gives x = e 2 , which is a real number. To verify, let x = e 2 . Then, by definition, ln ( x ) = ln ( e 2 ) = 2.

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Is the following true: log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? Verify the result.

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Is the following true: ln ( e 1.725 ) ln ( 1 ) = 1.725 ? Verify the result.

No; ln ( 1 ) = 0 , so ln ( e 1.725 ) ln ( 1 ) is undefined.

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Real-world applications

The exposure index E I for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation E I = log 2 ( f 2 t ) , where f is the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

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Refer to the previous exercise. Suppose the light meter on a camera indicates an E I of 2 , and the desired exposure time is 16 seconds. What should the f-stop setting be?

2

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The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 M 2 where M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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