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Chapter 7: Problem 125

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(4^{x+3}=\frac{2^{x}}{16}\)

Short Answer

Expert verified
x = -10

Step by step solution

01

Simplify the Equation

First, rewrite the equation using the same base for all terms. Recall that 4 can be written as \(2^2\) and 16 as \(2^4\). Rewriting the equation, we get: \[ (2^2)^{x+3} = \frac{2^x}{2^4} \]
02

Apply Exponent Rules

Simplify the exponents by applying the power rule \[ (a^m)^n = a^{m \cdot n} \]. This converts our equation to: \[ 2^{(2(x+3))} = 2^{x-4} \] or \[ 2^{(2x+6)} = 2^{x-4} \]
03

Equate Exponents

Since the bases are the same, we can set the exponents equal to each other: \[ 2x + 6 = x - 4 \]
04

Solve for x

Isolate x by first subtracting x from both sides: \[ 2x + 6 - x = -4 \] Simplify to get: \[ x + 6 = -4 \] Finally, isolate x by subtracting 6 from both sides: \[ x = -4 - 6 \] Therefore, \[ x = -10 \]

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Key Concepts

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

Solving Equations
Solving an equation means finding the value of the variable that makes the equation true. In our example, we want to find the value of x that satisfies the equation \( 4^{x+3} = \frac{2^x}{16}\).
We start by simplifying the equation and making the bases the same. This often helps us see a clearer path to the solution. Once we have similar bases, we can equate their exponents.
The steps involved include:
  • Rewriting the terms using common bases.
  • Applying exponent rules to simplify.
  • Equating the exponents when the bases are the same.
  • Solving for the variable using basic algebraic methods like addition, subtraction, and division.
By following these steps, solving complex-looking exponential equations becomes much more straightforward.
Exponent Rules
Exponent rules are essential tools in simplifying expressions and solving equations. There are several important exponent rules you should know:
  • Power Rule: \( (a^m)^n = a^{m \times n} \) - When raising a power to another power, multiply the exponents.
  • Product Rule: \( a^m \times a^n = a^{m+n} \) - When multiplying like bases, add the exponents.
  • Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \) - When dividing like bases, subtract the exponents.
In our problem, we used the power rule to simplify \( (2^2)^{x+3} \) into \( 2^{2(x+3)} \) or \( 2^{2x+6} \). Recognizing and applying the right exponent rules helps in transforming and simplifying equations quickly.
Base Conversion
Converting bases is a critical step in simplifying exponential equations. Often, working with the same base allows us to directly compare and manipulate exponents. Here's what you need to know about base conversion:
  • Identify Common Bases: Numbers like 4, 8, and 16 can be converted to powers of 2 (e.g., \( 4 = 2^2 \), \( 16 = 2^4 \)).
  • Rewrite Using Common Bases: Rewrite each term in the equation using the identified base. For example, \( 4^{x+3} \) becomes \( (2^2)^{x+3} \).
  • Simplify Using Exponent Rules: Apply the exponent rules to further simplify. For instance, \( (2^2)^{x+3} \) simplifies to \( 2^{2(x+3)} \) or \( 2^{2x+6} \).
By converting to a common base, we simplify the equation and make it easier to solve. As seen, this makes equating the exponents straightforward, as we did transforming \( \frac{2^x}{16} \) into \( \frac{2^x}{2^4} = 2^{x-4} \).

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Most popular questions from this chapter

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, April 21,2018 , in Sitka, Alaska, high tide occurred at 4: 51 AM (4.85 hours) and low tide occurred at 11:50 AM (11.83 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 10.03 feet, and the height of the water at low tide was -0.46 feet. (a) Approximately when did the next high tide occur? (b) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (c) Use the function found in part (b) to predict the height of the water at 3 PM.

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A resistor and an inductor connected in a series network impede the flow of an alternating current. This impedance \(Z\) is determined by the reactance \(X\) of the inductor and the resistance \(R\) of the resistor. The three quantities, all measured in ohms, can be represented by the sides of a right triangle as illustrated, so \(Z^{2}=X^{2}+R^{2}\). The angle \(\phi\) is called the phase angle. Suppose a series network has an inductive reactance of \(X=400\) ohms and a resistance of \(R=600\) ohms. (a) Find the impedance \(\underline{Z}\). (b) Find the values of the \(\operatorname{six}\) trigonometric functions of the phase angle \(\phi\).
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