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0.1 Logarithm concepts -- the logarithm explained by analogy to

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This module discusses another way of understanding logarithms by providing an analogy to roots.

The logarithm may be the first really new concept you’ve encountered in Algebra II. So one of the easiest ways to understand it is by comparison with a familiar concept: roots.

Suppose someone asked you: “Exactly what does root mean?” You do understand roots, but they are difficult to define. After a few moments, you might come up with a definition very similar to the “question” definition of logarithms given above. 8 3 size 12{ nroot { size 8{3} } {8} } {} means “what number cubed is 8?”

Now the person asks: “How do you find roots?” Well...you just play around with numbers until you find one that works. If someone asks for 25 size 12{ sqrt {"25"} } {} , you just have to know that 5 2 = 25 size 12{5 rSup { size 8{2} } ="25"} {} . If someone asks for 30 size 12{ sqrt {"30"} } {} , you know that has to be bigger than 5 and smaller than 6; if you need more accuracy, it’s time for a calculator.

All that information about roots applies in a very analogous way to logarithms.

Roots Logs
The question x a size 12{ nroot { size 8{a} } {x} } {} means “what number, raised to the a power, is x?” As an equation, ? a = x size 12{? rSup { size 8{a} } =x} {} log a x size 12{"log" rSub { size 8{a} } x} {} means “ a size 12{a} {} , raised to what power, is x size 12{x} {} ?” As an equation, a ? = x size 12{a rSup { size 8{?} } =x} {}
Example that comes out even 8 3 = 2 size 12{ nroot { size 8{3} } {8} =2} {} log 2 8 = 3 size 12{"log" rSub { size 8{2} } 8=3} {}
Example that doesn’t 10 3 size 12{ nroot { size 8{3} } {"10"} } {} is a bit more than 2 log 2 10 size 12{"log" rSub { size 8{2} } "10"} {} is a bit more than 3
Out of domain example − 4 size 12{ sqrt { - 4} } {} does not exist ( x 2 size 12{x rSup { size 8{2} } } {} will never give − 4 size 12{ - 4} {} ) log 2 ( 0 ) size 12{"log" rSub { size 8{2} } \( 0 \) } {} and log 2 ( − 1 ) size 12{"log" rSub { size 8{2} } \( - 1 \) } {} do not exist ( 2 x size 12{2 rSup { size 8{x} } } {} will never give 0 or a negative answer)

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
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John Reply
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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
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David Reply
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David
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emma Reply
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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
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Adjanou
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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
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
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Mohammed
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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, Logarithms. OpenStax CNX. Mar 22, 2011 Download for free at http://cnx.org/content/col11286/1.1
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