1. Laws (Properties) of Logarithms

2. 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

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

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

5. Change of Base Formula

6. 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

7. 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

8. 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

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

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

11. Rewrite the expression as a single logarithm.

12. Rewrite the expression as a single logarithm.

13. Evaluate the expression without using the Change of Base Formula.

14. Evaluate the expression using the Change of Base Formula