Logarithms and Exponents

• Coal Consumption [Plascencia, 01/13/1997]
  How long will coal reserves last if consumption increase at the rate of 3.1 percent per year?

• Exp, Log, and Ln Functions Explained [Ojanen, 7/31/1996]
  What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?

• Logarithmic Scales [Brian, 01/16/2002]
  Why might empirical scientific data be transformed to logarithmic data and presented on a logarithmic scale?

• Logarithms [Wetmores, 12/2/1995]
  What is a logarithm?

• Logarithms: History and Use [Temple, 7/12/1996]
  How do I explain why logarithms work to non-math-oriented people? Also, what is the history of the development of the concept of logarithm? Why is it called a "logarithm"?

• Logarithms' Relation to Exponents [Vid, 09/29/2001]
  I understand what logs are, and their relation to Euler's constant, but I don't understand why they are what they are.

• Logarithms with Unknown Basis [Juan, 12/12/2001]
  Provided only that, in some basis, log 2 = 0.607 and log 3 = 0.959, find a good value for log 5.

• Negative Exponents Explained [Alec, 12/29/2001]
  I have a problem with exponents like 5 to the negative second.

• Significant Non-Zero Digits [Sarah, 11/27/2001]
  How many significant digits are there in a number with no non-zero digits? Example: 00.000 Are there any?

• Decimal Exponents [Segal, 11/3/1994]
  Hope you'll help a graduate student trying to understand an example in an economics textbook. It involves food production and the equation is: Q = 100(K[superscript.8]L[superscript.2]). K involves machines and L involves labor. The diagram shows Q(output) as 13,800, L as 760, K as 90. I have been unable to figure out how to make the equation reveal those numbers.

• Richter magnitude problem [Friedman, 11/23/1994]
  The Richter magnitude, R, of an earthquake is given by R=0.67 log(0.37E)+1.46 where E is the energy in kW*h released by the earthquake. Show that if R increases by 1 unit, E increases by a factor of about 31.

• The Difference Between Log and Natural Log [Chan, 2/8/1995]
  What is the difference between log and natural log?

• Log Problem [Robicheau, 5/5/1995]
  Solve: 3(3^x)+9(3^-x) = 28.

• Hints About Logs [Donabedian, 5/17/1995]
  Do you know any hints on how to figure out logarithms?

• Is Doubling Exponential? [Fox, 7/17/1995]
  The amount of information storable in silicon has roughly doubled every year. Many authors have written articles about this "exponential growth rate." Is doubling exponential?

• Using a Rule of Logarithms [Jackson, 9/13/1995]
  I can't figure out this problem. Can you help? log(sub)2 3k+log(sub)2 5=log(sub)2 45.

• Logarithm Problem [Farnell, 9/15/1995]
  For the problem 2log(base 5) (3a+1) = log(base 5) 4 - 2log(base 5) (2a-1) our book has an answer of 3, but we still can't figure out how they got it; the book may be wrong but we would like to know for sure.

• Another Logarithm Problem [Farnell, 10/2/1995]
  A math class asks how to convert between bases in logarithms.

• An Identity for Solving a Logarithm Problem [Bill, 1/25/1996]
  I need to know the identity for solving this logarithm problem...

• Composite Functions Using Logarithms [Anonymous, 3/10/1996]
  Suppose f and g are functions defined by f(x) = x+2 and g(x) = x. Find all x > -2 for which: 3^[g(x)*logbase3 f(x)] = f(x).

• Integral of Ln x [Red, 3/31/1996]
  What is the integral of the natural log of x (ln x)?

• Logarithm [Claren, 4/6/1996]
  How do you solve this question: log base 2, of 156?

• Why does ln(-x) x>0 equal ln(x)+pi*i? [Tolbert, 6/5/1996]
  Could you please explain why the ln(-x) x>0 equals ln(x)+pi*i?

• Logs in Different Bases [Brasher, 6/11/1996]
  Solving equations with different base logs.

• Logarithm Proof [Turner, 6/18/1996]
  A proof involving logarithms of different bases.

• No Logarithm of a Negative Number [emsmdt, 8/6/1996]
  log(-5) = ?

• Logarithm Formulae [Johnson, 8/16/1996]
  What is an algorithm or longhand procedure for calculating the logarithm of a number?

• Dividing 1005! by 10^N [Colleen, 9/5/1996]
  Determine the largest integer N such that 1005! is divisable by 10^N.

• Population Growth Rate [Research, 9/5/1996]
  If a population increases from 10 million to 300 million in 10,000 years, what is the annual growth rate?

• 1005! / 10^n: A Number Puzzle [Connors, 9/11/1996]
  Find the largest positive integer, n, such that 1005! is divisible by 10 to the power of n.

• Equations with Logarithms [Osman, 04/18/1997]
  Find those values of 'm' for which this equation has three different solutions: x^3 - (3/2)x^2 + 5/2 = log (base 1/4) (m).

• Exponential Decay [Tequilla, 05/02/1997]
  After 69 days, a certain amount of Radium F had decayed to 1/sqrt(2) of its original mass. Determine the half-life of Radium F.

• Population of China [Julie, 05/13/1997]
  If the population growth rate of China is 4.3 percent and the current population of the country is 1.27 billion people, when will the population reach 2 billion?

• Logs in Equations [Coyle, 05/20/1997]
  How do you get rid of logs when you're trying to solve an equation?

• Exponentiation [Pomerance, 08/16/1997]
  How do I calculate x^y using only exp, ln, log, and the trigonometric functions?

• Simplify the Expression [Mummon, 08/21/1997]
  ... and eliminate any negative exponents: (4b)^1/2(8b^2/5).

• Exponents on a Scientific Calculator [Brownfield, 08/21/1997]
  My problem is that I am not using the EXP key correctly...

• Logarithmic Equivalent of the Inverse Hybolic Cosine Function [Gurnell, 10/28/1997]
  I recently attempted a question which I think I have done wrong and I was wondering if there were a set formula for this problem...

• Natural Logarithms [Waissblut, 11/01/1997]
  What's "natural" about natural logarithms? Why is 'e' a transcendental number?

• Proving e is Irrational [Decker, 11/19/1997]
  My professor suggested using a proof by contradiction, but I don't understand how to do it.

• Logarithims and Base E [Steven, 11/20/1997]
  Why is the base of a natural logarithim is "e" - how did "e" receive a value of 2.17... ?

• Logs with Different Bases [Kevin, 11/26/1997]
  How do you do an addition problem involving logs with different bases?

• Dividing Positive and Negative Exponents [Matt, 12/18/1997]
  Do I put the variables with negative exponents on the bottom?

• Natural Logs [Alison, 12/20/1997]
  I need to understand natural logs. Could you please explain them in the simplest terms possible?

• E in the Modern World [Vena, 01/06/1998]
  I am looking for examples of how e exists in the modern world.

• Algebra 2: Exponents [Wolf, 01/07/1998]
  I need to find out what "x" is in this problem: x^x^3 = 3.

• What Does x Equal? [Hoffman, 01/16/1998]
  32 * 4^2x = 12. What is x? What is a logarithm?

• Logarithms [Eveline, 01/25/1998]
  Why are there so many rules? How do you solve 4^x+3 = 7^x-1 ?

• Euler's Number [Holcomb, 01/28/1998]
  Why is Euler's number referred to as the natural base?

• 'Log' Button [Edey, 01/29/1998]
  We have been using the 'log' button on our calculators to solve problems involving compound interest. What does that button do?

• Logarithms: Solving for T [Hubble, 01/30/1998]
  3.5 = (1.12)^T. We can't remember how to solve for T

• Integration of Natural Logs [Jaeger, 02/12/1998]
  Why does the natural log of x equal the integral of 1/t dt from 1 to x? Why is INT[(1/t)dt] from 1 to x the natural log of x, or why was it defined this way?

• History and Applications of the Natural Logarithm [Fornwalt, 03/02/1998]
  I'm so surprised at how often the number e comes up. Where did it come from? Who first derived it? Why is it so common in the field of biology?

• Finding a Decimal Exponent [Tyler, 03/21/1998]
  How do you find 32^0.4 without using a calculator?

• How Does a Calculator Find Log? [Chuan, 03/23/1998]
  Is there no specific formula for deriving logarithms in decimal form?

• The Log of a Negative Number [Witty, 03/27/1998]
  My calculator can't do log (-3). Is the answer a complex number?

• Logarithms in Bases Other Than 10 [Tweedie, 03/28/1998]
  What methods are available for solving logarithms in a base other than 10?

• Finding Logarithms By Hand [Campbell, 04/10/1998]
  How would you find a logarithm without a calculator or a table?

• Applications of Logarithms [Heem, 04/15/1998]
  How can I calculate the logarithm function on a number without a calculator? What are some of the uses of the logarithm function?

• Converting Logs to Another Base [Suleman, 04/27/1998]
  How can I use the logarithm function in base e to find a logarithm in base 10 or a logarithm in base n?

• Solving Logistic Equations for a Variable in the Exponent [Cavenett, 05/03/1998]
  Using natural logarithms to solve for the time variable in two logisitic growth equations.

• Evaluating Large Numbers [Clifton, 06/11/1998]
  How do you evaluate 2^71 * 3? Can you use logarithms?

• Prove that Log A is Irrational [Magictek, 06/14/1998]
  Can you help me prove that a common log of a number (not powers of 10) is irrational?

• John Napier [Ricciardi, 06/17/1998]
  Why was John Napier's discovery of logarithms a great contribution to the field of mathematics?

• A Logarithmic Equation [Stratton, 08/06/1998]
  How do you solve A*2^(x/y) = B for x using a base 2 logarithm?

• Solving Logarithms with the Quotient Rule [Sparks, 11/11/1998]
  Can you help me solve this logarithm problem for x? I think you need to use the quotient law.

• Measurements on a Logarithmic Scale [Tori, 11/21/1998]
  What kinds of measurements are done on a logarithmic scale? What physical properties make a logarithmic scale more appropriate?

• Logarithmic Scales [Patel, 12/13/1998]
  Why do we use logarithmic scales and in what applications do they appear?

• Exponents Containing Negative Integers [Harrison, 12/15/1998]
  Order of operations and raising a negative number to a power.

• Rule of 72 [Vasseur, 01/26/1999]
  Why does the rule of 72 work? For what interest values does it work best?

• Applications of Non-antidifferentiable Functions [Mekisha , 01/29/1999]
  What are some of the real life applications of the functions sin (x)/ x, e^(-x^2), and sin (x^2)?

• Logarithms before 1965 [Nicole, 02/08/1999]
  Could you tell me how logarithms were used before the year 1965?

• (i)th Root and (i)th Power [Evan, 02/13/1999]
  How do you simplify x to the power of i (and 1/i), where x could be any number?

• Multiplication of Powered Numbers [Duane , 02/17/1999]
  What is the product of two numbers with different powers, and how many zeros are there at the end of the product?

• Fractional Exponents [Tom, 03/11/1999]
  I am trying to figure out the water consumption of the average goat.

• Slide Rules [Ben, 03/13/1999]
  Can you help me learn how to use a slide rule?

• Etymology of 'Logarithm' [Marco, 03/19/1999]
  How does the term 'logarithm' relate to the terms 'exponent' and 'knots', and to nautical logbooks?

• Decimal Exponents [Richard, 03/19/1999]
  Find x^y, where y is a decimal, without using a calculator.

• Srinivasa Ramanujan [Gee, 03/28/1999]
  What are the mathematical contributions of Srinivasa Ramanujan?

• Biology Logarithms [Shekman007, 04/01/1999]
  The Ph of an acidic pond is 5. What is the hydrogen ion concentration in moles per liter?

• Explicit Inverses [Yin, 04/21/1999]
  How do you solve x^x = 5 without graphing or guessing?

• Euler Equation and DeMoivre's Theorem [Anthony, 05/18/1999]
  Do you have a proof of the equation e^(i*Pi) + 1 = 0?

• Finding the Base of a Logarithm [Theodora, 06/08/1999]
  How do I solve log (base x) 0.05 = -0.5 for x?

• Polynomial Function and Exponential Growth [Roland, 06/08/1999]
  How can I find a polynomial function with integer coefficients that has 3, 2, -4 and -1 as zeroes? What is the population of a colony of bacteria 20 seconds after it is discovered?

• Rules of Exponents [Helton, 10/02/1999]
  Use the rules of indices to simplify the expression [(16x^2)^(1/4)* (4x^3)^(1/2)/[(27^4)^(1/3)].

• EE Key on a Calculator [Galczynski, 10/25/1999]
  What does "EE" on the calculator stand for?

• Natural Logarithms [Hinojosa, 11/15/1999]
  Why are natural logarithms called "natural"?

• Number of Ancestors 1000 Years Ago [Larman, 11/22/1999]
  How many living relatives would I have had 1,000 years ago, assuming 25 years per generation?

• Solving a Decibel Noise Equation [Hazelden, 05/23/2000]
  How do you rearrange the formula 50 = 94.5 - 20log(r) - 8 to solve for r?

• Base of Common Logarithms [Shannon, 05/23/2000]
  Why is the base of common logarithms 10? Is it easier to work with, or is there a mathematical explanation?

• Solving an Equation with Sines and Exponents [Boisse, 05/24/2000]
  How can I solve the equation sin^2(exp(c-x)) + sin^2(exp(c+x)) = 0, where x is a (real) unknown and c is a real positive constant?

• What is i^(Googol)? [Feest, 05/27/2000]
  What is the value of i raised to the googol power?

• The Symbol for Natural Log [Abrams, 06/28/2000]
  Why is "ln" and not "nl" the abbreviation for natural log?

• Proving a^x = a^y iff x = y [Lauren, 12/13/2000]
  How can I prove that a^x = a^y iff y = x for all real numbers x and y?

• Anti-Logarithms [Elisabeth, 02/27/2001]
  Could you please explain the concept of anti-logarithms?

• When FOIL Fails [Ryan, 03/22/2001]
  Find the product - (2c-3)3rd power. I can do FOIL on a question to the second power, but I cannot figure out how to do a question to the third power.

• Resolving Decimal Exponents [Adel, 03/26/2001]
  How can you find 7^.3 or 5^.6 without using a calculator? Is there more than one way of doing this?

• Half Life of Money [Davis, 04/13/2001]
  What is the standard accepted formula for determining the amount of time it takes for money to lose half of its value due to inflation? I've found two, they are: t = t = log2/r and t = log2/log(1+r).

• Definition of Logarithm [Hongki, 04/16/2001]
  In the definition of logarithm, a^x = b iff x = log_a(b) where a and b are greater than 0 and a is not equal to 1, why the stipulations? Are there any problems or contradictions that arise if they are not given?

• Finding Roots on a Calculator [Moore, 04/22/2001]
  How can I use the equation HPR = (1.667)^(1/5) on a calculator that does not have a fifth root key?

• Interpreting 2^3^2 [Douglas, 04/26/2001]
  Why is 2^3^2 interpreted as 2^(3^2) = 512 instead of (2^3)^2 = 64?

• Solving Logarithm Equations [Thomas, 04/29/2001]
  How can I solve the equation: log_7(x) / log_7(4) - log_7(y) = 1, for y?

• Zero to a Negative Exponent [Bailis, 05/06/2001]
  Is 0^(-3) equal to 0, or is it undefined? We can't determine whether to use the 0^x = 0 rule, or to interpret it as 1/(0^3).

• Speed of Light [Kyrja, 06/08/2001]
  What's m s^-1? Why isn't it just m/s (meters per second) instead? What's the little "-1" at the end?

• Converting mW to dB [Jeff, 06/14/2001]
  I found this equation during a radio receiver discussion: 4x10-12mW = -114dBm. How does it equate?

• 2/3 Power using a Calculator [Tan, 07/05/2001]
  Can -512 be the answer to the equation x^(2/3) = 64?

• i and (-1) with Multiple Powers [Kenneth, 08/09/2001]
  What is the order in which i and (-1) should be raised when using multiple powers? Why is it possible to obtain so many different values?

• Antilogs [Cedric, 08/22/2001]
  If log sub(a)10 = 0.250, then log sub(10)a equals ? The solution requires me to the take the base(a) antilogarithm of both sides. That would be 10 = a^0.250. Why is this the final answer?

• 2^99 Divided by 99 [Sanjeev, 08/27/2001]
  What is the remainder when you divide 2^99 by 99?

• Briggs on Logarithms [Khalid, 08/28/2001]
  I want to know how Briggs constructed logarithmic tables of common logarithms.

• Factoring: Difference of Cubes [Brian, 09/05/2001]
  Factor a^3b^3 - 8x^6y^9.

• Briggs Logarithms [Khalid, 09/06/2001]
  Why did Briggs raise 2 to the tenth power? Why did he extract the square root of 1.024 instead of 1024? Why did he extract the square root of 1.024 forty seven times?...

• Solve for x: 131,072 = 2^x [Niki, 09/12/2001]
  How do I solve the following equation for x: 131,072 = 2^x? I know the answer is 17, but can't remember how to solve it.

• Binary Exponentiation [Abe, 09/13/2001]
  Show how to compute 5^21 using 6 multiplications. How many multiplications are needed to compute 7^7024?

• Base of an Exponential Function [Stefanie, 09/15/2001]
  Why can't the base of an exponential function be negative?

• n Factorial - Prove Lower Bound is n^(n/2) [Johnny, 09/11/2001]
  I am trying to prove the following: n^(n/2) <= n!

• Scientific Notation and Engineering Notation [Mark, 10/10/2001]
  Convert 0.000 000 000 000 000 000 000 3 to: a) scientific notation; b) engineering notation.

• Decimal Exponents [Alex, 11/07/2001]
  Can you have exponents that are decimals?

• Numbers Raised to the Negative Power [Rosemary, 11/14/2001]
  I know that 5^(-N) = 1/5^N. I would like to know why.

• Exponents as Variables [Bart, 11/20/2001]
  I can solve these problems but I don't know how to prove them: 9^1+x= 27; 2^-x-4=1/32; 243=(1/3)^x+4; 64=0.5)^3-x.

• Finding 13^99 [Potter, 11/21/2001]
  What is the units digit of 13 to the 99th power?

• Negative Numbers to Powers [Filoxus, 11/23/2001]
  If u is an irrational number and x is a negative number, what is x^u? How do I determine even whether that number is positive or negative?

• Manually Calculating Logs and Exponents [Wedge, 12/16/2001]
  Can you give me a formula to calculate base 10 logarithms and exponents?

• Logarithmic Scales [Brian, 01/16/2002]
  Why might empirical scientific data be transformed to logarithmic data and presented on a logarithmic scale?

• Logarithm Equation [Timmy, 01/22/2002]
  Is there a way to solve this equation: 2x^2-1 = ln(2x^2) ?

• Converting a Base 2 Log [Sue, 01/31/2002]
  2147483647 is one less than what power of 2?

• Are Exponents Associative? [Lucas, 02/05/2002]
  How Much Is 2^(i) or X^(i) ?

• Importance of Linear Functions [Abreeka, 02/06/2002]
  What is the importance of linear functions in the real world, and how are they used?

• Limits Approaching Infinity [Miranda, 02/16/2002]
  Is the only way to answer them by memorizing all the general questions you may come across?

• Using the Distributive Property [Melissa, 03/06/2002]
  This question is actually from my chemistry class: -2.4222 x 10^-19 - (-6.0556 x 10^-20).

• Base e, In, Log [Jerry, 03/08/2002]
  Problems in base e. For example: 3e^(2x-1) = 7, or e^(x+1) = 8.

• Doubling Time [Ariele, 03/08/2002]
  If the cost of an all-day ski lift ticket at Colorado's Vail Ski Area has been growing at about 7% per year ever since 1963, when a ticket was $5, what did it cost in 1993?

• Algorithm for Simple Fractional Exponents. [Matthew, 03/14/2002]
  How can I explain to my program how to resolve simple fractional exponents using only addition, subtraction, multiplication, division, square, and square root?

• Scientific Notation: Adding and Subtracting Exponents [Vicki, 03/21/2002]
  How do you add and subtract numbers written in scientific notation form?

• Logarithms of the Zeros of a Quadratic [Mary, 03/24/2002]
  If p and q are the zeros of 2x^2-mx+1=0, what is the value of log(base 2)p + log(base 2)q?