Laws (Properties) of Logarithms

Objectives Rewrite (expand) a logarithm to contain no logarithms of products, quotients, or powers. Rewrite (condense) a logarithmic expression into the logarithm of a single value. Use the change of base formula to rewrite a logarithm as a common logarithm or a natural logarithm. Basically, use the log laws to simplify, expand, or evaluate

Vocabulary product rule: multiplication becomes addition quotient rule: division becomes subtraction power rule: exponent becomes multiplier

Laws of Logarithms product rule - multiplication becomes addition quotient rule - division becomes subtraction power rule - exponent becomes multiplier

Change of Base Formula

Proof for the Product Rule loga xy = loga x + loga y Proof: Step 1:  Let m = logax and n = logay Step 2: Write in exponent form  x = am and y = an Step 3: Multiply x and y x y = am an = am+n Step 4: Take log a of both sides and evaluate  log a xy = log a am+n log a xy = (m + n) log a a log a xy = m + n log a xy = loga x + loga y

Proof for the Quotient Rule logax/y  = logax - logay Proof: Step 1:  Let m = logax and n = logay Step 2: Write in exponent form  x = am and y = an Step 3: Divide x by y x ÷ y = am ÷ an = am – n Step 4: Take log a of both sides and evaluate  loga (x ÷ y) = logaam - n loga(x ÷ y) = (m - n) logaa loga(x ÷ y) = m - n loga(x ÷ y) = logax - logay

Proof for the Power Rule loga xn = nloga x Proof: Step 1:  Let m = logax Step 2: Write in exponent form  x = am Step 3: Raise both sides to the power of n xn = ( am )n Step 4: Convert back to a logarithmic equation logaxn = mn Step 5: Substitute for m = logax logaxn = n logax

Use the Laws of Logarithms  to rewrite the expression in a form with no logarithm of a product, quotient or power.

Use the Laws of Logarithms  to rewrite the expression in a form with no logarithm of a product, quotient or power.

Rewrite the expression as a single logarithm.

Rewrite the expression as a single logarithm.

Evaluate the expression without using the Change of Base Formula.

Evaluate the expression using the Change of Base Formula