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  # Log Properties: Complete Lesson for Beginners
  
  Before entering residential properties of logarithms, let’s briefly discuss the connection between logarithms and exponents. The logarithm of a number is specified as t the power or index to increase a given base to acquire it. Let’s understand more about log properties with examples below.
  
  Considered that ax = M; where an and M is more than zero and also a ≠ 1, then we can symbolically represent this in logarithmic kind as;
  
  log a M = x.
  
  Examples:
  
  10-2= 1/100 = 0.01 ⇔ log 1001 = -2.
  
  3– 4= 1/34 = 1/81 ⇔ log 3 1/81 = -4.
  
  70= 1 ⇔ log 7 1 = 0.
  
  54= 625 ⇔ log 5 625 = 4.
  
  32= 9 ⇔ log 3 9 = 2.
  
  26= 64 ⇔ log 2 64 = 6.
  
  10-2= 0.01 ⇔ log 1001 = -2.
  
  ### Logarithmic Properties or Log Properties
  
  Logarithm homes and also policies work because they allow us to broaden, condense or solve logarithmic formulas. It for these reasons.
  
  Equipment of Linear Equations – Solve Equipments of Equations.
  
  ×.
  
  In most cases, you are informed about memorizing the guidelines when fixing logarithmic issues, yet, exactly how are these rules acquired.
  
  In this post, we are to head to consider the residential or commercial properties, and regulations of logarithms are derived using the rules of exponents.
  
  ### Product properties of logarithms.
  
  The product regulation states that multiplication of two or even more logarithms with usual bases is equal to including the private logarithms i.e.
  
  log a (MN) = log a M + log a N.
  
  Proof.
  
  Allow x = log aM and y = log a.
  
  Convert each of these formulas to the rapid kind.
  
  ⇒ a x = M.
  
  ⇒ a y = N.
  
  Multiply the rapid terms (M & N):.
  
  ax * ay = MN.
  
  Considering that the base is joint, for that reason, including the backers.
  
  a x + y = MN.
  
  Taking log with base ‘a’ on both sides.
  
  log a (a x + y) = log a (MN).
  
  Using power guideline of a logarithm.
  
  Log a Mn ⇒ n log a M.
  
  ( x + y) log a = log a (MN).
  
  ( x + y) = log a (MN).
  
  Currently, replace the values of x and y in the formula we get above.
  
  log a M + log a N = log a (MN).
  
  For this reason, proved.
  
  log a (MN) = log a M + log a N.
  
  #### Examples:
  
  log50 + log 2 = log100 = 2.
  
  log 2 (4 x 8) = log 2 (22 x 23) =5.
  
  Quotient residential or commercial property of logarithms.
  
  This regulation mentions that the proportion of two logarithms with the same bases is equal to the difference of the logarithms i.e.
  
  log a (M/N) = log a M– log a N.
  
  Proof.
  
  Allow x = log aM and y = log a.
  
  Convert each of these formulas to the exponential type.
  
  ⇒ a x = M.
  
  ⇒ a y = N.
  
  Split the exponential terms (M & N):.
  
  ax/ ay = M/N.
  
  Because the base is joint, for that reason, deduct the exponents.
  
  a x– y = M/N.
  
  Taking log with base ‘a’ on both sides.
  
  log a (a x– y) = log a (M/N).
  
  Applying power regulation of logarithm on both sides.
  
  Log a Mn ⇒ n log a M.
  
  ( x– y) log a = log a (M/N).
  
  ( x– y) = log a (M/N).
  
  Currently, replace the worths of x as well as y in the formula we obtain above.
  
  log a M– log a N = log a (M/N).
  
  For this reason, it is verified.
  
  log a (M/N) = log a M– log a N.
  
  ### Power log properties
  
  According to the power building of logarithm, the log of a number’M’ with exponent ‘n’ amounts to the item of backer with the log of a number (without backer) i.e.
  
  Hence, log a M n = n log a M.
  
  Proof.
  
  Let’s assume,
  
  x = log a M.
  
  Now, let’s rewrite it as an exponential equation.
  
  a x = M.
  
  On both sides of the equation, take power ‘n.’
  
  ( a x) n = M n.
  
  ⇒ a xn = M n.
  
  Take visit both sides of the formula with the base a.
  
  Log a xn = log a M n.
  
  log a xn = log a M n ⇒ xn log a = log a M n ⇒ xn = log a M n.
  
  Currently, replace the values of x and y in the formula we get above.
  
  We know.
  
  x = log a M.
  
  So.
  
  xn = log a M n ⇒ n log a M = log a M n.
  
  Thus, confirmed.
  
  Log a M n = n log a M.
  
  ### Examples:
  
  log1003 = 3 log100 = 3 x 2 = 6.
  
  ## Adjustment of base log properties
  
  According to the modification of base building of logarithm, we can rewrite a given logarithm as the ratio of 2 logarithms with any brand-new base. It is given as:
  
  log a M = log b M/ log b N.
  
  Or.
  
  log a M = log b M × log N b.
  
  Its evidence can be done using one to one home and power rule for logarithms.
  
  Proof.
  
  Express each logarithm in rapid kind by allowing.
  
  Let.
  
  x = log N M.
  
  Transform it to exponential form.
  
  M = N x.
  
  Apply one to one home.
  
  log b N x = log b M.
  
  Using the power guideline.
  
  X log b N = log b M.
  
  Isolating x.
  
  x = log b M/ log b N.
  
  Replacing the value of x.
  
  log a M = log b M/ log b N.
  
  Or we can write it as.
  
  log a M = log b M × log a b.
  
  Thus, verified.