Simplifying Logarithmic Expressions Without Using Tables

The expression given is not clear due to the lack of parentheses and the use of ambiguous notation. Specifically, we need to know the bases of all the logarithms and the arguments for each term. However, for the sake of demonstration, let us assume the following expression:

3(log332/3) - 2(log39) (log33/8)

Step 1: Clarify the Expression

To ensure clarity, we should clearly define the expression using proper notation. The corrected expression would be:

3 log3(32/3) - 2 log3(9) log3(3/8)

Step 2: Apply Logarithmic Properties

Before simplifying, let's review some important logarithmic properties:

logb(MN) logbM logbN logb(M/N) logbM - logbN logb(Mn) n logbM

Step 3: Simplify Each Term

Let's simplify each term one by one.

Term 1: 3 log3(32/3)

Using the property: logb(M/N) logbM - logbN

3 log3(32/3) 3(log332 - log33)

Simplify further:

3(log332 - 1) 3(log332 - log33) 3 log3(32/3)

Term 2: 2 log3(9)

Using the property: logb(Mn) n logbM

2 log3(9) 2 log3(32) 2(2 log33) 4 log33

Since log33 1, we have:

4 log33 4(1) 4

Term 3: log3(3/8)

Using the property: logb(M/N) logbM - logbN

log3(3/8) log33 - log38 1 - log38

Further simplification:

log38 log3(23) 3 log32

Thus, log3(3/8) 1 - 3 log32

Step 4: Combine All Terms

Now, we can combine all the simplified terms:

3 log3(32/3) - 2 log3(9) log3(3/8) 3(log332 - 1) - 4 (1 - 3 log32)

Simplify the expression:

3(log332 - 1) 3 log332 - 3

Thus, the final expression is:

(3 log332 - 3) - 4 1 - 3 log32

Simplify further:

3 log332 - 6 - 3 log32

Combine like terms:

3 log332 - 3 log32 - 6

Conclusion

In conclusion, the simplified expression is:

3 log332 - 3 log32 - 6

This can be further simplified if we know the value of log32, but for now, it is in its simplified form.

Related Keywords

logarithmic expressions simplification logarithms algebra base