Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 64-\log 16 $$

Step-by-Step Solution

Answer

The value is approximately 0.60.

Step 1: Review the properties of logarithms

Recall the properties of logarithms, specifically the difference property: \( \log_a M - \log_a N = \log_a \left(\frac{M}{N}\right) \). We can utilize this property to simplify the original expression.

Step 2: Apply the Logarithm Difference Rule

Using the difference property of logarithms, rewrite the expression: \( \log 64 - \log 16 = \log \left(\frac{64}{16}\right) \). This simplifies the computation by combining the logarithms into a single logarithm.

Step 3: Simplify the fraction inside the logarithm

Next, simplify the fraction inside the logarithm: \( \frac{64}{16} = 4 \). This reduces the logarithmic expression to \( \log 4 \).

Step 4: Calculate \( \log 4 \) to the nearest hundredth

Using a calculator, find the value of \( \log 4 \). Remember, this denotes the common logarithm (base 10). The approximate value of \( \log 4 \) is 0.602. Ensure to round it to the nearest hundredth as required.

Key Concepts

Difference Property of Logarithms

The difference property of logarithms is a powerful tool. It allows us to simplify expressions involving the subtraction of two logarithms. This property states that the difference between two logarithms with the same base can be rewritten as a single logarithm of a fraction. The formula is:

• \( \log_a M - \log_a N = \log_a \left(\frac{M}{N}\right) \)

This means the logarithm of a quotient (fraction) is the difference of the logarithms. Imagine you have the expression \( \log 64 - \log 16 \). Using the difference property, you rewrite it as \( \log \left(\frac{64}{16}\right) \). This significantly simplifies calculations.
By understanding this property, you can quickly transform and simplify complex logarithmic expressions into a more manageable form.

Logarithmic Simplification

In mathematics, simplifying expressions is key to solving problems efficiently. When dealing with logarithms, the goal is often to reduce an expression to its simplest form using fundamental properties like the difference property. Once we apply this property, the next step is simplifying the fraction within the logarithm to make calculations straightforward.

• For instance, in our earlier example: \( \frac{64}{16} = 4 \).

Breaking down the fraction helps you transform the problem into an easily computable form: \( \log 4 \).
Logarithmic simplification reduces the complexity, makes the math less intimidating, and set you up for the final computation step. This set of skills not only aids in manual calculations but boosts your confidence in handling logarithms.

Common Logarithm

A common logarithm is simply a logarithm with base 10, often written as \( \log \). It's one of the most frequently used logarithmic functions because of its simplicity and easy calculation on standard calculators. Whenever you see \( \log \) without a base, it is understood to be base 10.

• For example, finding \( \log 4 \) means you're working with \( \log_{10} 4 \).

Using a calculator, you compute \( \log 4 \) resulting in an approximate value of 0.602. Remember to round to the nearest hundredth if expressed in a school problem or real-world task. Common logarithms make it easier for people, especially students, to grasp and manipulate exponential relationships. This accessibility is why calculators often default to base 10 when computing logarithms.