/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Solve. See Example 4. $$ \lo... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 57

Solve. See Example 4. $$ \log _{4} 16=x $$

Short Answer

Expert verified
The solution is \( x = 2 \).

Step by step solution

01

Understanding the Logarithm Equation

The given equation is \( \log_{4} 16 = x \). This means we are trying to find the value of \( x \) such that when 4 is raised to the power of \( x \), it equals 16. In other words, \( 4^x = 16 \).
02

Rewrite 16 in Terms of Base 4

To solve \( 4^x = 16 \), it is helpful to express 16 as a power of the base 4. Since 16 can be written as \( 4^2 \) (because \( 4 \times 4 = 16 \)), the equation becomes \( 4^x = 4^2 \).
03

Equate the Exponents

Since the bases are the same (both are 4), we can set the exponents equal to each other to solve for \( x \). Thus, \( x = 2 \).
04

Solution Conclusion

With the solution \( x = 2 \), we have confirmed that raising the base 4 to the power 2 yields 16, which satisfies the original logarithm equation.

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Key Concepts

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

Base Conversion
Base conversion is the process of changing a number from one base to another. In the context of logarithms, it simplifies problems by expressing numbers in terms of a common base. Take the equation \( \log_{4} 16 = x \). To solve this, you can convert 16 into a power of 4, because 4 is the base of the logarithm. To find out:
  • Determine what exponent the base needs to be raised to become the target number.
  • For instance, 16 can be written as \( 4^2 \) because \( 4 \times 4 = 16 \).
Recognizing relationships between numbers makes solving equations easier. This understanding of base conversion helps resolve logarithmic and exponential problems quickly.
Exponentiation
Exponentiation involves raising a base number to the power of an exponent. In \( 4^x = 16 \), 4 is the base, and \( x \) is the exponent. Exponentiation answers questions like: "How many times must 4 be multiplied by itself to make 16?"Understanding Exponential Notation:
  • The base (4) is multiplied by itself \( x \) times.
  • If you express 16 as \( 4^2 \), you see that \( x \) must be 2 to satisfy this equation.
Exponentiation is vital in math, helping in everything from solving equations to understanding growth patterns. Mastering exponent manipulations can simplify logarithmic operations as seen in this example.
Logarithmic Equations
Logarithmic equations involve using logarithms to solve for unknown variables. In \( \log_{4} 16 = x \), the task is to find the exponent \( x \) such that 4 raised to \( x \) equals 16.To break it down:
  • A logarithm measures how many times a number (base) must be multiplied by itself to reach another number.
  • In this example, \( 4^x = 16 \) suggests rewriting 16 as \( 4^2 \).
  • Then, match the exponents to find \( x \). Here, \( x = 2 \).
By converting and manipulating logarithmic expressions, you solve for unknowns. Logarithmic equations like this one are common in math, and understanding them helps tackle more complex problems efficiently.

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Most popular questions from this chapter

Graph. $$ y=\left(\frac{1}{3}\right)^{|x|} $$

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Solve each equation for \(x\). Give an exact solution and a four-decimal-place approximation. See Examples 3 and 7. $$ \log x=1.3 $$

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