/*! 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 21 Solve each equation. $$ 5^{x... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 21

Solve each equation. $$ 5^{x}=\frac{1}{125} $$

Short Answer

Expert verified
x = -3

Step by step solution

01

Recognize the base

Identify the base number in the given equation. Here, the base number is 5.
02

Express the equation with the same base

Rewrite the right-hand side of the equation \(\frac{1}{125}\) using the base 5. We know that \(\frac{1}{125} = 5^{-3}\), since \({125 = 5^{3}}\). Thus, the equation becomes \(5^x = 5^{-3}\).
03

Set the exponents equal

Since the bases are the same, set the exponents equal to each other: \(x = -3\).

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Key Concepts

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

Exponents
Understanding exponents is crucial for solving equations involving powers. Exponents tell us how many times a base is multiplied by itself. For example, in the expression \(5^3\), the base is 5 and the exponent is 3. This means \(5 \times 5 \times 5\), which equals 125. Here are some key points to remember about exponents:
  • Any number raised to the power of 1 is the number itself, e.g., \(a^1 = a\).
  • Any number raised to the power of 0 is 1, e.g., \(a^0 = 1\).
  • Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent, e.g., \(a^{-n} = \frac{1}{a^n}\).
With this understanding, recognizing different exponent forms can help solve exponential equations with ease.
Base Identification
In exponential equations, identifying the correct base is an essential step. The base is the number that is being raised to a power. For example, in \(5^x = \frac{1}{125}\), the base is 5. Correct base identification helps in simplifying and transforming the equation for easier solving. Key points to remember:
  • Look for a common base on both sides of the equation. This can simplify the process significantly.
  • If the equation involves complex numbers, try to break them down into prime factors, aiding base identification.
In our example, recognizing that 125 is actually \(5^3\) allows us to write \(\frac{1}{125}\) as \(5^{-3}\). Now, both sides of the equation have the same base, allowing us to proceed to the next step.
Equation Solving
Once the bases in an exponential equation are the same, solving the equation becomes straightforward. You can set the exponents equal to each other and solve for the unknown variable. Here are the steps in our example:
  • We have \(5^x = 5^{-3}\).
  • Since the bases are the same (both are 5), we can set the exponents equal: \(x = -3\).
  • Solve for \(x\) to get the final answer: \(x = -3\).
Always remember that the key to solving such equations is having the same base on both sides so you can easily compare the exponents.

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Most popular questions from this chapter

Which equation defines a one-to-one function? Explain why the others are not, using specific examples. A. \(f(x)=x\) B. \(f(x)=x^{2}\) C. \(f(x)=|x|\) D. \(f(x)=-x^{2}+2 x-1\)

In the central Sierra Nevada of California, the percent of moisture that falls as snow rather than rain is approximated reasonably well by $$f(x)=86.3 \ln x-680$$ where \(x\) is the altitude in feet. (a) What percent of the moisture at 5000 ft falls as snow? (b) What percent at 7500 ft falls as snow?

If the function is one-to-one, find its inverse. \(f(x)=3 x+1\)

If the function is one-to-one, find its inverse. \(f(x)=x^{3}+5\)

If \(f(x)=4^{x},\) find each value indicated. In Exercise 50, use a calculator, and give the answer to the nearest hundredth. \(f\left(-\frac{1}{2}\right)\)
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