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Chapter 4: Problem 63

Simplify the expression. $$ 5^{\log _{5}(x+y)} $$

Short Answer

Expert verified
The simplified expression is \( x+y \).

Step by step solution

01

Understand the Logarithm Rule

Recall that if you have an expression of the form \( a^{ \text{log}_a (b)} \), it simplifies to \( b \). This is because the base \(a\) and the logarithm base \(a\) essentially cancel each other out.
02

Apply the Rule

Given the expression \( 5^{\text{log}_5(x+y)} \), identify that it matches the form \( a^{ \text{log}_a (b)} \) with \( a = 5 \) and \( b = x+y \).
03

Simplify

Using the rule, simplify the expression \( 5^{\text{log}_5(x+y)} \) to \( x+y \).

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Key Concepts

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

logarithm properties
Logarithms are powerful tools in mathematics that help in solving equations involving exponentials. One of the most useful properties of logarithms is their ability to 'cancel out' with exponentials of the same base. For any positive base \( a \) different from 1 and any positive number \( b \), the expression \( a^{ \text{log}_a (b)} \) simplifies directly to \( b \). This property arises because logarithms are the inverse operations of exponentials. Think of it like folding a piece of paper to its original size; the logarithmic operation 'undoes' the exponential operation, streamlining complex expressions.
exponentials
Exponential functions are expressions where a constant base is raised to a variable exponent, like \( 5^x \). They grow very quickly compared to linear or polynomial functions. The rule \( a^{ \text{log}_a (b)} = b \) utilizes the nature of exponential and logarithmic functions being inverses of each other. For example, in the logarithmic expression \( 5^{\text{log}_5(x+y)} \), the base 5 and the logarithm with base 5 effectively neutralize each other. Thus, what remains is \( x + y \), simplifying the entire expression effortlessly. This property allows for easier manipulation in many algebraic contexts.
algebraic expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. Understanding the interplay between different types of expressions, including logarithmic and exponential forms, is key to mastering algebra. For instance, to simplify \( 5^{\text{log}_5(x+y)} \), you recognize it as a specific algebraic form where the logarithm's base matches the base of the exponent. The core of simplifying such expressions is recognizing patterns and applying rules correctly. This transforms the seemingly complex \( 5^{\text{log}_5(x+y)} \) into the much simpler \( x+y \), making the solution far more straightforward.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(3^{6 x+5}=5^{2 x}\)

Graph the following functions on the window [-3,3,1] by [-1,8,1] and comment on the behavior of the graphs near $$ \begin{array}{l} x=0 \\ \mathrm{Y}_{1}=e^{x} \\ \mathrm{Y}_{2}=1+x+\frac{x^{2}}{2} \\ \mathrm{Y}_{3}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6} \end{array} $$

Write \(10^{2 x-4}=80,600\) in logarithmic form.

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(1024=19^{x}+4\)

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{4}(3 d)+\log _{4} 1=\log _{4}(3 d) $$
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