Problem 25
Question
Use the properties of logarithms to write the expression as a single logarithm. $$ \log _{7}(x)+\log _{7}(x-3)-2 $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \( \log_7\left(\frac{x^2 - 3x}{49}\right) \).
1Step 1: Combine using Product Rule
The logarithmic property \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \) allows us to combine the two logarithms. Apply this to \( \log_7(x) + \log_7(x-3) \) to get \( \log_7(x(x-3)) \).
2Step 2: Simplify the Expression
Multiply inside the logarithm: \( \log_7(x(x-3)) = \log_7(x^2 - 3x) \).
3Step 3: Express Subtraction as a Logarithm Division
The property \( \log_b(M) - c = \log_b\left(\frac{M}{b^c}\right) \) means we can write the expression \( \log_7(x^2 - 3x) - 2 \) as \( \log_7\left(\frac{x^2 - 3x}{7^2}\right) \).
4Step 4: Simplify the Denominator
Compute the power to simplify the expression further: \( \log_7\left(\frac{x^2 - 3x}{49}\right) \) by evaluating \( 7^2 = 49 \).
Key Concepts
Product RuleLogarithmic PropertiesSimplifying ExpressionsDivision Property of Logarithms
Product Rule
The product rule is a fundamental property of logarithms that helps simplify the addition of logs. It's similar to multiplying in algebra, but here we're multiplying inside a logarithm.
When you have two logarithms with the same base added together, like \( \log_b(M) + \log_b(N) \), you can combine them into a single logarithm: \( \log_b(M \cdot N) \).
This makes our calculations much smoother and reduces the number of separate logs. In the given exercise, you have \( \log_7(x) + \log_7(x-3) \).
Applying the product rule here, this becomes \( \log_7(x(x-3)) \). By understanding this rule, you can transform complex logarithmic expressions into simpler forms.
When you have two logarithms with the same base added together, like \( \log_b(M) + \log_b(N) \), you can combine them into a single logarithm: \( \log_b(M \cdot N) \).
This makes our calculations much smoother and reduces the number of separate logs. In the given exercise, you have \( \log_7(x) + \log_7(x-3) \).
Applying the product rule here, this becomes \( \log_7(x(x-3)) \). By understanding this rule, you can transform complex logarithmic expressions into simpler forms.
Logarithmic Properties
Logarithmic properties are powerful tools for simplifying lengthy expressions. They include rules like the product rule, quotient rule, and power rule.
These rules help us break down complex logs into actionable parts.
In our original example, the focus was on combining and reducing terms.
By using the product rule, we transformed a sum into a product, making the log easier to handle. Then, using knowledge of other properties, like the division property of logarithms, we continue to simplify.
These properties allow us to manipulate logs in algebraic ways, making it easier to solve equations or condense expressions.
These rules help us break down complex logs into actionable parts.
In our original example, the focus was on combining and reducing terms.
By using the product rule, we transformed a sum into a product, making the log easier to handle. Then, using knowledge of other properties, like the division property of logarithms, we continue to simplify.
These properties allow us to manipulate logs in algebraic ways, making it easier to solve equations or condense expressions.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form where no further reduction is possible.
For logarithms, this usually means applying various properties to get rid of any extra terms or coefficients.
In the original exercise, after using the product rule, the expression inside the log \( \log_7(x(x-3)) \) became \( \log_7(x^2 - 3x) \).
This is a simpler form where multiplication has been carried out. Simplifying expressions aids in making them more straightforward for further calculations or interpretations.
For logarithms, this usually means applying various properties to get rid of any extra terms or coefficients.
In the original exercise, after using the product rule, the expression inside the log \( \log_7(x(x-3)) \) became \( \log_7(x^2 - 3x) \).
This is a simpler form where multiplication has been carried out. Simplifying expressions aids in making them more straightforward for further calculations or interpretations.
Division Property of Logarithms
The division property of logarithms allows us to deal with subtraction in logarithmic expressions.
Just as with multiplication, subtraction in logs can be transformed using this property:\( \log_b(M) - c = \log_b\left(\frac{M}{b^c}\right) \).
This transformation helps us express a subtraction of a constant as a division inside the logarithm. In our solution, after calculating \( \log_7(x^2 - 3x) - 2 \), you can apply the division property to get \( \log_7\left(\frac{x^2 - 3x}{49}\right) \). This step eases calculations and problem-solving by converting a multi-component expression into a single, more manageable logarithmic form.
Just as with multiplication, subtraction in logs can be transformed using this property:\( \log_b(M) - c = \log_b\left(\frac{M}{b^c}\right) \).
This transformation helps us express a subtraction of a constant as a division inside the logarithm. In our solution, after calculating \( \log_7(x^2 - 3x) - 2 \), you can apply the division property to get \( \log_7\left(\frac{x^2 - 3x}{49}\right) \). This step eases calculations and problem-solving by converting a multi-component expression into a single, more manageable logarithmic form.
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