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Chapter 4: Problem 4

Solve for \(x\).\(4^{x}=\frac{1}{256}\)

Short Answer

Expert verified
The solution to the exponential equation \(4^x = 1/256\) is \(x = -4\).

Step by step solution

01

Express \(1/256\) as a Power of 4

Observe that \(1/256 = 4^{-4}\). Both 4 and 256 are powers of 2, which enables us to check this by calculating \(4^{-4} = (2^2)^{-4} = 2^{-8} = 1/256\). Thus the original equation \(4^x = 1/256\) can be rewritten as \(4^x = 4^{-4}\).
02

Compare the Exponents

Now that both sides of the equation are expressed with the same base, we can set the exponents equal to each other: \(x = -4\).
03

Check the Solution

To validate this solution, we can insert \(x = -4\) into the original equation and check if both sides are equal: \(4^{-4} = 1/256\), which is true. Thus, the solution is valid.

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Key Concepts

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

Powers of Numbers
When dealing with powers of numbers, we use an exponent to indicate how many times a number, called the base, is multiplied by itself. For instance, in the expression \(4^x\), the number 4 is the base and \(x\) is the exponent. The exponent tells us the number of times to use the base in a multiplication. Here's a simple breakdown:
  • If \(x = 2\), then \(4^x = 4^2 = 4 \times 4 = 16\).
  • If \(x = 3\), then \(4^x = 4^3 = 4 \times 4 \times 4 = 64\).
It's important to remember the pattern: as the exponent increases, the result (also known as the power) gets rapidly larger, showcasing exponential growth.
Negative Exponents
In cases like \(4^{-4}\), negative exponents can initially be confusing, but they actually provide a clear mathematical purpose. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. That transforms \(4^{-4}\) into \(\frac{1}{4^4}\). Unlike positive exponents that denote repeated multiplication, negative exponents tell us to divide:
  • \(4^1 = \frac{1}{4^{-1}}\) or \(4\)
  • \(4^2 = \frac{1}{4^{-2}}\) or \(16\)
  • Finally, using \(4^{-4}\), we get \(\frac{1}{256}\)
Negative exponents are a useful tool that helps in simplifying complex expressions and solving exponential equations effortlessly.
Equivalent Expressions
Understanding equivalent expressions is crucial for solving equations like \(4^x = \frac{1}{256}\). Equivalent expressions refer to different expressions that have the same value for all values of the variable. In our problem, expressing \(\frac{1}{256}\) as \(4^{-4}\) allows direct comparison with \(4^x\).Why is this useful?
  • Allows simplification of equations: By having identical bases, we can equate the exponents, reducing a complex problem to a simpler form.
  • Improves accuracy: Working with one base reduces complexity and eliminates chances for error.
It's all about transforming the equation into a form that is easier to analyze, ultimately leading us to find that \(x = -4\) is the correct and equivalent solution.

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Most popular questions from this chapter

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(2^{x}-7=0\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.0825}{26}\right)^{26 t}=9\)

Super Bowl Ad Cost The table shows the costs \(C\) (in millions of dollars) of a 30 -second TV ad during the Super Bowl for several years from 1987 to \(2006 .\) (Source: TNS Media Intelligence)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Cost } \\ \hline 1987 & 0.6 \\ \hline 1992 & 0.9 \\ \hline 1997 & 1.2 \\ \hline 2002 & 2.2 \\ \hline 2006 & 2.5 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1987 .\) (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use a graphing utility to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to predict the costs of a 30 -second ad during the Super Bowl in 2009 and in 2010 .

Solve for \(y\) in terms of \(x\).\(\ln y=\ln (2 x+1)+\ln 1\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.065}{365}\right)^{365 t}=4\)
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