# Evaluate Logarithms Using Properties Worksheet

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Evaluating logarithms using properties involves applying key rules to simplify expressions: the Product Rule ($$\log_b(MN) = \log_b(M) + \log_b(N)$$), the Quotient Rule ($$\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$$), and the Power Rule ($$\log_b(M^k) = k \cdot \log_b(M)$$). These properties are effective only when the bases are the same. It facilitates breaking down complex logarithmic expressions into simpler, more manageable parts, making it easier to solve logarithmic equations and perform calculations.
Example: Simplify $$\log_2(8 \cdot 4) - \log_2(16) + \log_2(2^3)$$.

Algebra 2
Logarithms

## How Will This Worksheet on "Evaluate Logarithms Using Properties" Benefit Your Student's Learning?

• Using these properties, students can break down and simplify complicated logarithmic problems, making them easier to solve.
• Applying these rules helps students think critically and logically, which improves their overall problem-solving skills.
• Knowing these properties is key for higher-level math classes, like calculus and advanced algebra, where these concepts come up often.
• Using the product, quotient, and power rules makes logarithmic calculations faster and more efficient.
• Working with logarithmic properties helps students get better at algebra, especially with skills like factoring and combining terms.

## How to Evaluate Logarithms Using Properties?

• Use the logarithmic properties to combine the expression into a single logarithm, often using the quotient rule to express the difference between logarithms and the product rule to express the addition of logarithm.
• Simplify the resulting logarithmic expression by performing any necessary arithmetic inside the logarithm.
• If needed, rewrite any numbers inside the logarithm to match the base, using exponentiation to simplify further.
• If the expression is still not simplified, then again use properties of logarithm and simplify the expression to its final form.

## Solved Example

Q. Use properties of logarithms to evaluate the expression. $\log_5 5 + \log_5 25$
Solution:
1. Sum of logs: Sum of logs: $\log_5 5 + \log_5 25$$\newline$ Use product property: $\log_b P + \log_b Q = \log_b (PQ)$
2. Apply product property: Apply product property: $\log_5 5 + \log_5 25 = \log_5 (5 \cdot 25)$
3. Simplify inside log: $\newline$ $\log_5 (5 \cdot 25) = \log_5 125$
4. Express $125$ as power: $\newline$$125 = 5^3$$\newline$ So, $\log_5 125 =\log_5 (5^3)$
5. Use power property:$\newline$$\log_b (P^Q) = Q \cdot \log_b P$$\newline$ So, $\log_5 (5^3) = 3 \cdot \log_5 5$
6. Evaluate: Evaluate: $3 \times\log_5 5$$\newline$ Since, $\log_5 5 = 1$.$\newline$ $3 \times\log_5 5 = 3$

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