# [[Combining Logarithms with Same Base and Argument]] (Mathematics > Algebra) ## Definition When two or more logarithmic terms have the same base and argument, they can be combined by adding or subtracting their coefficients. This property allows for simplifying expressions involving logarithms with identical bases and arguments. For example, given: $ a \log_b c + d \log_b c $ The terms can be combined as: $ (a + d) \log_b c $ ## Key Concepts - Only logarithms with the same base and argument can be combined. - Coefficients are added or subtracted depending on the signs of the terms. - Combining logarithms simplifies expressions and makes calculations easier. ## Important Properties 1. The base and the argument must be identical for the terms to be combined. 2. Coefficients of the logarithmic terms are simply added or subtracted. 3. The logarithmic rule applies to expressions of the form $a \log_b c$. ## Essential Formulas - Combining logarithms with the same base and argument: $ a \log_b c + d \log_b c = (a + d) \log_b c $ ## Core Examples 1. **Basic Example**: Simplify $5 \log_4 7 + 3 \log_4 7$. $ 5 \log_4 7 + 3 \log_4 7 = (5 + 3) \log_4 7 = 8 \log_4 7 $ 2. **Advanced Example**: Simplify $2 \log_5 3 - \log_5 3$. $ 2 \log_5 3 - \log_5 3 = (2 - 1) \log_5 3 = \log_5 3 $ ## Related Theorems/Rules - [[Logarithmic Properties]]: Includes the product, quotient, and power rules for logarithms. ## Common Pitfalls - Trying to combine logarithms with different bases or arguments. - Forgetting to add or subtract the coefficients correctly. ## Related Topics - [[Logarithmic Rules]] - [[Logarithmic Expansion]] ## Quick Review Questions 1. How would you simplify $4 \log_2 5 + 6 \log_2 5$? 2. Can you combine $3 \log_3 8 + 5 \log_4 8$? Why or why not? *** ![[Combining_Logarithms_with_Same_Base_and_Argument_visualization]]