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Chapter 5: Problem 6

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

Short Answer

Expert verified
The solution is \(x = \frac{1}{3}\).

Step by step solution

01

Understand the Equation

The given equation is \(125^{x} = 5\). The goal is to solve for \(x\).
02

Rewrite the Base

Recognize that 125 and 5 can be expressed as powers of 5. We know that 125 is \(5^3\), so we rewrite 125 as \((5^3)^x = 5\).
03

Apply the Power Rule

Using the power rule \((a^m)^n = a^{m imes n}\), we can rewrite \((5^3)^x\) as \(5^{3x}\). This gives us the equation \(5^{3x} = 5^1\).
04

Equate the Exponents

Since the bases are the same, equate the exponents: \(3x = 1\).
05

Solve for x

Divide both sides of the equation by 3 to solve for \(x\):\[ x = \frac{1}{3} \]
06

Conclusion

Therefore, the solution to the equation \(125^x = 5\) is \(x = \frac{1}{3}\).

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Key Concepts

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

Base Conversion
When solving exponential equations, a common approach is base conversion. It makes handling these equations much easier. Consider the original problem: \(125^{x} = 5\). Here, we recognize that both 125 and 5 share a common base, which is 5. By converting the numbers into this base, the equation becomes simpler.
  • 125 can be rewritten as \(5^3\) because multiplying 5 by itself three times (\(5 \times 5 \times 5\)) results in 125.
  • Similarly, 5 is already in its simplest form, \(5^1\).

Using base conversion allows us to compare the powers directly, facilitating the solution process. Recognizing common bases early on can significantly streamline solving these equations.
Exponent Rules
Exponent rules are essential for working with and solving exponential equations. In this problem, understanding these rules helps significantly in simplifying expressions.

One important rule is the power of a power rule: \((a^m)^n = a^{mn}\). This rule states that when you raise a power to another power, you can multiply the exponents:
  • In our equation, \((5^3)^x\) is transformed to \(5^{3x}\), meaning we multiply 3 by \(x\).
  • This step turns the complex expression into something much more manageable, \(5^{3x} = 5^1\).

By applying the exponent rules correctly, solving the equation becomes straightforward after simplifying both sides to the same base.
Solving Equations
Solving exponential equations often involves simplifying and equating exponents. Once you've used base conversion and exponent rules, like in our example, you simplify the original problem down to an equation of the form \(5^{3x} = 5^1\).

Here, solving becomes easier when both sides of the equation have the same base. You can then directly set the exponents equal to each other:
  • This gives you the equation \(3x = 1\), based on matching the exponents from both sides.

Finally, solve for \(x\) by dividing both sides by 3, yielding \(x= \frac{1}{3}\). The process involves breaking down complex exponential equations into simpler arithmetic operations, ensuring a smooth path to the solution.

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Most popular questions from this chapter

The magnitude of a star is defined by the equation $$M=6-2.5 \log \frac{I}{I_{0}}$$ where \(I_{0}\) is the measure of a just-visible star and \(I\) is the actual intensity of the star being measured. The dimmest stars are of magnitude \(6,\) and the brightest are of magnitude 1. Determine the ratio of light intensities between a star of magnitude 1 and a star of magnitude 3.

The concentration of bacteria \(B\) in millions per milliliter after \(x\) hours is given by $$B(x)=1.33 e^{0.15 x}$$ (a) How many bacteria are there after 2.5 hours? (b) How many bacteria are there after 8 hours? (c) After how many hours will there be 31 million bacteria per milliliter?

Suppose \(f(r)\) is the volume (in cubic inches) of a sphere of radius \(r\) inches. What does \(f^{-1}(5)\) represent?

The following table shows the average Valentine's Day spending in dollars per consumer for various years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2011 & 2012 & 2013 \\\ \hline \text { Spending ( } \$ \text { ) } & 103 & 116 & 126 & 131 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate values for \(a\) and \(b\) so that \(f(x)=a+b \ln x\) models the data, where \(x=1\) corresponds to \(2010, x=2\) to 2011 and so on. (b) Use your function to estimate average spending in 2012 and compare to the value in the table.

Escherichia coli is a strain of bacteria that occurs naturally in many organisms. Under certain conditions, the number of bacteria present in a colony is approximated by $$A(t)=A_{0} e^{0.023 t}$$ where \(t\) is in minutes. If \(A_{0}=2,400,000,\) find the number of bacteria at each time. Round to the nearest hundred thousand. (a) 5 minutes (b) 10 minutes (c) 60 minutes
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