logarithm question

Question

Write log_8??x^2/(yz^2 )? as a sum, difference, or product of logarithms. Simplify the result if possible

Answer 1

Daniel,

 

Let me make sure I understand the question correctly:  You need to write the following logarithm as a sum, difference, or product of logs...

 

log₈  (x²)/(y*z²)

 

Before I get to answering your question, let me review the log rules quickly:

1. Product rule:  log (a*b) = log (a) + log (b)
2. Division rule:  log (a/b) = log (a) - log (b) (ORDER MATTERS!)
3. Exponent rule:  lob (a²) = log (a)² = 2*log (a)  (This is NOT the same as (log a)²!!!)

Now that we have those rules, let's apply them to this question.  Start with the math operation that is connecting all parts of (x²)/(y*z²) together: the division sign.

Apply Rule #2:

log₈ (x²)/(y*z²)   -->  log₈ (x²) - log₈ (y*z²)

Now we have two separate logs to work with.  Let's focus on log₈ (x²) first.

Apply Rule #3:

log₈ (x²)   -->   2*log₈ (x)

Now for the second part: log₈ (y*z²).

Apply Rule #1:

log₈ (y*z²)   -->   log₈ (y) + log₈ (z²)

If you notice, this part can be simplified further.  Apply Rule #3 to the log containing z².

log₈ (y*z²) --> log₈ (y) + log₈ (z²)   -->   log₈ (y) + 2*log₈ (z)

Now that everything has been simplified down as far as we can go, let's update the whole expression:

log₈ (x²)/(y*z²) --> log₈ (x²) - log₈ (y*z²)   -->   2*log₈ (x) - [log₈ (y) + 2*log₈ (z)]

If you're comparing this answer to the answers in the back of a textbook, they may take this one step further and distribute the subtraction sign through to the log₈(y) and the 2log₈(z), in which case, your answer would look like this:

2*log₈ (x) - log₈ (y) - 2*log₈ (z)

Hope this is helpful!!

~Patty