Problem 15 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$5^{3 \log _{5} 2}$$

Step-by-Step Solution

Answer

The expression simplifies to 8.

Step 1: Recognize the Property

Understand that the expression can be simplified by using the property of logarithms. Specifically, the property \( a^{ ext{log}_a(b)} = b \) can be applied here.

Step 2: Identify the Base and Exponent

Notice that the base of the power is 5, and the exponent is \(3 \log_{5}(2)\). The expression can thus be written as \( (5^{\log_{5}(2)})^3 \).

Step 3: Apply the Exponent Property

Using the property \( a^{\text{log}_a(b)} = b \), simplify \( 5^{\log_{5}(2)} \) to 2. The expression now becomes \( 2^3 \).

Step 4: Calculate the Exponentiation

Calculate the power by multiplying 2 by itself three times: \( 2 \times 2 \times 2 = 8 \). Thus, \( 2^3 \) simplifies to 8.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves two elements: a base and an exponent. It essentially tells you how many times you need to multiply the base by itself. Let's break down these components:

• **Base**: The number that you are going to multiply.
• **Exponent**: The number that indicates how many times the base is used in the multiplication.

For example, in the expression \(5^3\), the base is 5, and the exponent is 3. This means you'll multiply 5 by itself three times, resulting in \(5 \times 5 \times 5 = 125\). This operation forms the core of our exercise where understanding how exponentiation works allows us to simplify expressions efficiently. It's crucial to remember that exponentiation is not the same as multiplication, as it's a repeated operation focusing on powers rather than sums.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics that allows you to reduce the complexity of mathematical phrases to their simplest form. This process uses various mathematical properties to consolidate and minimize the elements of an expression.

In our particular exercise, we utilized the property \(a^{\log_a(b)} = b\) to simplify a complex-looking expression. This property indicates that if your base and the logarithmic base match, the expression inside the log can be extracted directly.

Let's apply this: Given an expression \(5^{3 \log_5 (2)}\), we can split it as \((5^{\log_5 (2)})^3\). Recognizing that \(5^{\log_5 (2)}\) simplifies directly to 2 (using our property), the expression then turns into \(2^3\), which is much easier to compute.

Logarithmic Identities

Logarithmic identities are rules that simplify the use of logarithms in mathematical expressions. They are helpful for solving complex logarithmic problems and can transform them into more manageable forms.

One critical logarithmic identity is \(a^{\log_a(b)} = b\), which we used extensively in solving our exercise. This identity relies on matching the base of the logarithm and the base of the exponent. When they match, the expression simplifies significantly because the exponent effectively cancels out the base of the logarithm.

Another useful identity is the change of base formula, which can convert logarithms from one base to another but it was not necessary in this problem.

• These identities simplify equations and support the comprehension of logarithmic operations.
• Mastering them streamlines solving and understanding exponential and logarithmic expressions.

By applying the appropriate logarithmic identity, you can efficiently solve expressions without needing complex computation or calculators.