Problem 20 - Precalculus with Calculus Previews

Question

In Problems \(19-22\), find the exact value of the given expression. $$ 25^{\log _{5} 8} $$

Step-by-Step Solution

Answer

The exact value is 64.

Step 1: Understand the Expression

We need to find the exact value of the expression \(25^{\log_{5} 8}\). This expression is in the form \(a^{\log_{b} c}\), which can be simplified using the change of base formula.

Step 2: Simplify the Base

First, note that \(25 = 5^2\). So we can rewrite the expression as \((5^2)^{\log_{5} 8}\). This helps in using the properties of exponents to simplify the problem.

Step 3: Apply Exponentiation Rules

According to the power of a power property in exponents, \((a^m)^n = a^{m \cdot n}\). Apply this by multiplying the exponents: \((5^2)^{\log_{5} 8} = 5^{2 \cdot \log_{5} 8}\).

Step 4: Simplify Using Logarithmic Identity

We can use the identity \(a^{\log_{a} b} = b\) to simplify. Here, we have \(5^{2 \cdot \log_{5} 8} = 5^{\log_{5} (8^2)} = 8^2\), because 2 can be moved inside as a power of the argument in the logarithm.

Step 5: Calculate the Final Answer

Now calculate \(8^2\), which is \(64\). Thus, the exact value of the expression \(25^{\log_{5} 8}\) is \(64\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation used to denote repeated multiplication. In simple terms, when you raise a base number to an exponent, you're multiplying that base by itself as many times as the exponent specifies. For instance, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent, implying \(a\) is multiplied by itself \(n\) times.
Exponentiation has several applications, especially in fields that require calculations at varying scales. It simplifies expressions and calculations, transforming complex multiplication sequences into a singular representation. An essential aspect of exponentiation is recognizing patterns and rules that help simplify expressions, such as the properties of exponents.

Properties of exponents

Understanding the properties of exponents is crucial for simplifying and solving expressions involving powers. Here are some key properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\). Multiply powers with the same base by adding their exponents.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). Raise a power to another power by multiplying the exponents.
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) when \(aeq 0\). Divide powers with the same base by subtracting their exponents.
• Zero Exponent: \(a^0 = 1\) for any non-zero \(a\). Any number raised to the power zero equals one.
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\). Negative exponents represent reciprocals of the base raised to the positive opposite exponent.

These properties simplify many mathematical problems and are especially useful in algebra and calculus. Recognizing and applying these rules reduces complex expressions into more manageable forms.

Change of base formula

The change of base formula is a handy tool when working with logarithms of different bases. Often, calculators and certain mathematical tools are optimized for base 10 logarithms (common logarithm) or base \(e\) (natural logarithm). The change of base formula allows us to convert any logarithm to a base more convenient for calculation or comparison.
The formula is expressed as:\[\log_b a = \frac{\log_c a}{\log_c b}\]where \(\log_b a\) is the logarithm of \(a\) with base \(b\), and \(c\) is the new base to which you're converting. This is particularly useful in situations like this problem, where converting between logarithms of different bases aids simplification. Utilizing this formula helps overcome limitations when the desired base is not directly available.

Logarithmic identities

Logarithmic identities are powerful tools that help simplify logarithmic expressions and solve complex equations. Here are some key identities to remember:

• Product Rule: \(\log_b (xy) = \log_b x + \log_b y\). The logarithm of a product equals the sum of the logarithms.
• Quotient Rule: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\). The logarithm of a quotient is the difference between the logarithms.
• Power Rule: \(\log_b (x^r) = r \cdot \log_b x\). Bringing powers down in a log allows multiplication of the outside exponent.
• Base Change: \(b^{\log_b x} = x\). This identity is pivotal in simplifying expressions, as it transforms the log back into a simple term.

These identities enable transformations and simplifications, making logarithmic expressions more accessible. When solving problems like \(25^{\log_5 8}\), using these identities turns a seemingly complex problem into straightforward calculations.