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Chapter 12: Problem 77

Use the special properties of logarithms to evaluate each expression. \(8^{\log _{8} 5}\)

Short Answer

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5

Step by step solution

01

Understand the Given Expression

Before evaluating, note that the given expression is in the form of a logarithmic exponential function: \(8^{\log_{8} 5}\).
02

Apply the Property of Logarithms

Recall the logarithmic identity: \(a^{\log_{a} b} = b\). The base \(a\) raised to the logarithm with the same base simplifies directly to the argument of the logarithm.
03

Simplify the Expression

Using the property from Step 2, we simplify \(8^{\log_{8} 5}\) as follows: \(8^{\log_{8} 5} = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Identity
In this section, we will explore a fundamental rule related to logarithms. The logarithmic identity is an important property of logarithms that states: \(a^{\log_{a} b} = b\). This identity means that when the base of an exponent and the base of the logarithm are the same, the expression simplifies to the argument of the logarithm. Let's break it down with an example: If you have \(8^{\log_{8} 5}\), you can directly simplify it to聽5 because both the exponent and the logarithm share the base (which is 8 in this instance). This property allows us to handle complex logarithmic expressions easily and is particularly useful in algebra and calculus.
Exponential Functions
Exponential functions involve expressions of the form \(a^{x}\). Here, \(a\) is the base, and \(x\) is the exponent. Exponential functions grow rapidly because the variable \(x\) appears in the exponent. They are used widely in science, engineering, and finance to model growth and decay. For instance, in population growth models, if the base is greater than one, the function represents rapid growth. In our given problem, understanding exponential functions helps in recognizing patterns like \(8^{\log_{8} 5}\). Knowing that exponents and logarithms are inverses of each other is critical for simplifying these expressions correctly.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions often requires the use of properties like the logarithmic identity. Let's revisit our example: \(8^{\log_{8} 5}\). By recognizing that the base of the exponent (8) matches the base of the logarithm, you apply the property \(a^{\log_{a} b} = b\), and simplify the expression directly to 5. Other useful properties include:
  • Product Rule: \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\)
  • Quotient Rule: \(\log_{a}(b/c) = \log_{a}(b) 鈭� \log_{a}(c)\)
  • Power Rule: \(\log_{a}(b^{c}) = c \cdot \log_{a}(b)\)
These rules are vital tools when working with logarithms. They allow us to break down complex expressions into simpler components, making them easier to solve.

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Most popular questions from this chapter

Determine whether each statement is true or false. $$\log _{3} 7+\log _{3} 7^{-1}=0$$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$3 \log _{a} 5-\frac{1}{2} \log _{a} 9$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 2^{19}$$

Solve each equation. \(\log _{6} \sqrt{216}=x\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{2 x}=4 $$
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