Problem 64
Question
Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log x y^{2} z^{3}\)
Step-by-Step Solution
Verified Answer
\( \log x + 2 \cdot \log y + 3 \cdot \log z \)
1Step 1: Use the properties of logarithms
In this step, we apply the property that the logarithm of a product is the sum of the logarithms of the factors. This property states: \( \log(a \cdot b) = \log a + \log b \). So our expression becomes: \( \log x + \log(y^{2}) + \log(z^{3}) \).
2Step 2: Use the power rule for logarithms
Now we apply the power rule for logarithms, which states that \( \log(a^{b}) = b \cdot \log a \). Apply this to the terms \( \log(y^{2}) \) and \( \log(z^{3}) \). This gives us: \( \log x + 2 \cdot \log y + 3 \cdot \log z \). This is the simplified expression.
Key Concepts
Logarithm of a ProductPower Rule for LogarithmsSimplifying Logarithmic Expressions
Logarithm of a Product
Logarithms help us transform complex expressions into easier ones. This is particularly true when working with products, thanks to the logarithm of a product property.
When you see a product inside a logarithm like \(\log(xyz)\), you can break it down using the rule: \(\log(a \cdot b) = \log a + \log b\). This means the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Applying this property helps simplify the expression into smaller, more manageable parts. If you have more than two factors, like in the example \(\log(xy^{2}z^{3})\), just keep applying the rule:
The idea is to break down complex logs into simpler parts, making calculations easier.
When you see a product inside a logarithm like \(\log(xyz)\), you can break it down using the rule: \(\log(a \cdot b) = \log a + \log b\). This means the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Applying this property helps simplify the expression into smaller, more manageable parts. If you have more than two factors, like in the example \(\log(xy^{2}z^{3})\), just keep applying the rule:
- First, separate it into \(\log x + \log(y^{2}z^{3})\).
- Next, further simplify by splitting \(\log(y^{2}z^{3})\) into \(\log(y^{2}) + \log(z^{3})\).
The idea is to break down complex logs into simpler parts, making calculations easier.
Power Rule for Logarithms
The power rule for logarithms is a super helpful tool when you come across exponents inside a logarithm. It follows the rule: \(\log(a^{b}) = b \cdot \log a\).
This rule allows you to take the exponent and "bring it down" in front of the logarithm, turning multiplication inside into a manageable form.
Let's see how this works with our previous example after applying the product property:
By using the power rule, you rewrite logarithms with exponents efficiently. This not only simplifies calculations but provides clarity in expressions by eliminating high powers inside logs.
This rule allows you to take the exponent and "bring it down" in front of the logarithm, turning multiplication inside into a manageable form.
Let's see how this works with our previous example after applying the product property:
- From \(\log(y^{2}) + \log(z^{3})\), we use the power rule.
- For \(\log(y^{2})\), it becomes \(2 \cdot \log y\).
- For \(\log(z^{3})\), it becomes \(3 \cdot \log z\).
By using the power rule, you rewrite logarithms with exponents efficiently. This not only simplifies calculations but provides clarity in expressions by eliminating high powers inside logs.
Simplifying Logarithmic Expressions
Once you've applied the product property and the power rule, you're close to simplifying logarithmic expressions. Simplification is the process of making expressions as straightforward as possible.
In our example, after applying both prior properties, we moved from \(\log x + \log(y^{2}) + \log(z^{3})\) to \(\log x + 2 \cdot \log y + 3 \cdot \log z\).
This final result represents the simplest form because there are no more products or powers within the logs.
Why simplify? Simplifying helps in comparing, evaluating, or integrating logs in broader problems. It's also crucial in engineering, sciences, and finance where logs often describe growths or changes.
The key is to apply each property step-by-step to transform complicated logarithmic expressions into easier forms. This approach not only helps solve problems efficiently but also aids in developing a better understanding of logarithms and their roles in mathematics.
In our example, after applying both prior properties, we moved from \(\log x + \log(y^{2}) + \log(z^{3})\) to \(\log x + 2 \cdot \log y + 3 \cdot \log z\).
This final result represents the simplest form because there are no more products or powers within the logs.
- Every factor is separated.
- Every power is managed.
Why simplify? Simplifying helps in comparing, evaluating, or integrating logs in broader problems. It's also crucial in engineering, sciences, and finance where logs often describe growths or changes.
The key is to apply each property step-by-step to transform complicated logarithmic expressions into easier forms. This approach not only helps solve problems efficiently but also aids in developing a better understanding of logarithms and their roles in mathematics.
Other exercises in this chapter
Problem 64
Solve for \(x\). See Example 3 . $$ \log _{12} x=0 $$
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Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. \(f(x)=-\frac{1}{3}
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Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=-3.71 $$
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Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 7^{x^{2}}=10 $$
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