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Chapter 12: Problem 69

Use the properties of logarithms to find each of the following. $$\log _{5}(125 \cdot 625)$$

Short Answer

Expert verified
\(\log_5(125 \cdot 625) = 7\)

Step by step solution

01

Use the Product Property of Logarithms

The product property of logarithms states that \(\log_b(M \cdot N) = \log_b(M) + \log_b(N)\). Therefore, apply this property to the given problem: \(\log_{5}(125 \cdot 625) = \log_{5}(125) + \log_{5}(625)\).
02

Express 125 and 625 as Powers of 5

Write 125 and 625 as powers of 5. We know that \(125 = 5^3\) and \(625 = 5^4\). Substitute these values back into the logarithm equation: \(\log_{5}(125) + \log_{5}(625) = \log_{5}(5^3) + \log_{5}(5^4)\).
03

Use the Power Property of Logarithms

The power property of logarithms states that \(\log_b(M^n) = n \log_b(M)\). Therefore, apply this property to the obtained logarithms: \(\log_{5}(5^3) = 3 \log_{5}(5)\) and \(\log_{5}(5^4) = 4 \log_{5}(5)\).
04

Simplify the Logarithms

Since \(\log_{5}(5) = 1\), simplify the expressions: \(3 \log_{5}(5) = 3 \cdot 1 = 3\) and \(4 \log_{5}(5) = 4 \cdot 1 = 4\).
05

Add the Results

Add the simplified results from Step 4: \(3 + 4 = 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

product property of logarithms
Logarithms are incredibly helpful when dealing with multiplication inside a log expression. The product property of logarithms states that \(\text{log}_b(M \times N) = \text{log}_b(M) + \text{log}_b(N)\). In other words, if you're taking the logarithm of a product, you can split it into the sum of two logarithms. For instance, with the expression \(\text{log}_5(125 \times 625)\), we can transform it into \(\text{log}_5(125) + \text{log}_5(625)\). This makes calculations much simpler and is the first important step in breaking down complex logarithmic expressions.
power property of logarithms
Another useful property of logarithms is the power property, which states that \(\text{log}_b(M^n) = n \text{log}_b(M)\). This means if you have an exponent inside the log, you can bring it out front as a coefficient. For example, using \(\text{log}_5(5^3)\) and \(\text{log}_5(5^4)\), we apply this property to get \(\text{log}_5(5^3) = 3 \text{log}_5(5)\) and \(\text{log}_5(5^4) = 4 \text{log}_5(5)\). This step is essential because it simplifies the expression to a more manageable form. Remembering that \(\text{log}_5(5) = 1\) helps reduce these expressions to simpler numeric values.
simplifying logarithmic expressions
Simplifying logarithmic expressions often involves using the properties of logarithms we've discussed. For example, after applying the product and power properties, you might end up with expressions like \(\text{log}_5(5^3)\) and \(\text{log}_5(5^4)\). When simplified further, these become \(\text{log}_5(5^3) = 3\) and \(\text{log}_5(5^4) = 4\). As \(\text{log}_5(5) = 1\), multiplying by 1 doesn鈥檛 change the value. Finally, we add these results: 3 + 4 = 7. Thus, \(\text{log}_5(125 \times 625) = 7\). This step-by-step simplification process makes tackling complex logarithmic problems more straightforward.

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