Math Problem Statement

log base 3 power 4 + log base 3 power 15 - log base 3 power 20

Solution

To solve the expression:

log34+log315log320\log_3 4 + \log_3 15 - \log_3 20

we can use the properties of logarithms. Specifically:

  1. The product rule: logba+logbc=logb(ac)\log_b a + \log_b c = \log_b (a \cdot c).
  2. The quotient rule: logbalogbc=logb(ac)\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right).

Step 1: Combine the first two terms

Using the product rule on log34+log315\log_3 4 + \log_3 15:

log34+log315=log3(415)=log360\log_3 4 + \log_3 15 = \log_3 (4 \cdot 15) = \log_3 60

Step 2: Apply the quotient rule

Now, we substitute this result into the original expression:

log360log320=log3(6020)\log_3 60 - \log_3 20 = \log_3 \left( \frac{60}{20} \right)

Step 3: Simplify the fraction

6020=3\frac{60}{20} = 3

So, the expression becomes:

log33\log_3 3

Step 4: Evaluate log33\log_3 3

Since log33=1\log_3 3 = 1 (because 31=33^1 = 3), we find:

log34+log315log320=1\log_3 4 + \log_3 15 - \log_3 20 = 1

Final Answer

1\boxed{1}

Would you like further details on logarithmic properties or have any questions?

Here are some related questions to explore:

  1. What is the result of log525+log54log510\log_5 25 + \log_5 4 - \log_5 10?
  2. How would the answer change if we used a different base for the logarithms?
  3. How does the change of base formula apply to logarithmic expressions?
  4. What happens when the expression includes square roots or exponents in logarithmic terms?
  5. Can we simplify expressions like logab+logaclogad\log_a b + \log_a c - \log_a d in general form?

Tip: Remember, combining logarithmic terms by multiplication or division can simplify complex logarithmic expressions!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Properties

Formulas

Product Rule for Logarithms: log_b(a) + log_b(c) = log_b(a * c)
Quotient Rule for Logarithms: log_b(a) - log_b(c) = log_b(a / c)

Theorems

Logarithmic Properties

Suitable Grade Level

Grades 10-12