Problem 52 - Algebra and Trigonometry

Question

\(45-54\) . Use the Laws of Logarithms to combine the expression. $$ 2\left(\log _{5} x+2 \log _{5} y-3 \log _{5} z\right) $$

Step-by-Step Solution

Answer

Combine using laws: \( \log_5 \left(\frac{x^2 y^4}{z^6}\right) \).

Step 1: Apply the Logarithm Multiplication Rule

The rule states that \( a \log_b(M) = \log_b(M^a) \). Here, since the entire expression is multiplied by 2, distribute the 2 to each term inside the parenthesis: \[ 2(\log_5 x + 2 \log_5 y - 3 \log_5 z) = \log_5 x^2 + 2\log_5 y^2 - 3\log_5 z^2 \] This yields \( \log_5 (x^2) + \log_5 (y^4) - \log_5 (z^6) \).

Step 2: Apply the Logarithm Addition Rule

The logarithm addition rule states that \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \). Apply this rule to combine the first two terms: \[ \log_5 (x^2) + \log_5 (y^4) = \log_5 (x^2 \cdot y^4) \] This simplifies the expression to \( \log_5 (x^2 y^4) - \log_5 (z^6) \).

Step 3: Apply the Logarithm Subtraction Rule

The logarithm subtraction rule states that \( \log_b(M) - \log_b(N) = \log_b(\frac{M}{N}) \). Apply this to combine the terms into a single logarithm:\[ \log_5 (x^2 y^4) - \log_5 (z^6) = \log_5 \left(\frac{x^2 y^4}{z^6}\right) \] Now the expression is fully combined into one logarithm.

Key Concepts

Logarithm Multiplication Rule

The logarithm multiplication rule is a handy tool to simplify expressions involving logs. It states that if you have a constant multiplied by a logarithm, such as \( a \log_b(M) \), you can rewrite this as \( \log_b(M^a) \). Here's how it works:

• The constant "a" becomes the exponent of the number inside the logarithm.
• This transformation helps move the constant into the logarithm itself.

For example, in the expression \( 2(\log_5 x + 2 \log_5 y - 3 \log_5 z) \), we distribute the "2" between each term. This turns into \( \log_5(x^2) + \log_5(y^4) - \log_5(z^6) \). See how each term now contains its exponent, making it simpler for further steps!

Logarithm Addition Rule

The logarithm addition rule allows us to combine two logarithms with the same base, transforming an addition into a single term. This rule states that \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \). It's easy to understand:

• When adding logs of the same base, multiply their arguments.
• This simplification results in fewer terms, converting multiple logs into one.

In our example, \( \log_5(x^2) + \log_5(y^4) \) becomes \( \log_5(x^2 \cdot y^4) \). Now, instead of two separate logarithms, you have one combined term, prepared for the next operation. This step reduces complexity and makes further calculations straightforward.

Logarithm Subtraction Rule

The logarithm subtraction rule simplifies expressions where one logarithm is subtracted from another with the same base. This rule looks like this: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \). Here's the rundown:

• Subtracting two logs is equivalent to dividing their arguments.
• This greatly condenses the expression, merging two logs into one.

For the expression \( \log_5(x^2 y^4) - \log_5(z^6) \), we apply the subtraction rule to get \( \log_5\left(\frac{x^2 y^4}{z^6}\right) \). Now all terms are merged neatly into a single logarithm, achieving simplicity and clarity in the expression.