/*! 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 16 Solve each equation. $$ 8^{x... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 16

Solve each equation. $$ 8^{x}=64 $$

Short Answer

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

Step by step solution

01

Express Both Sides with the Same Base

Recognize that 64 can be written as a power of 8. Specifically, 64 is the same as 8 squared. Therefore, rewrite the equation as \(8^{x} = 8^{2}\).
02

Set the Exponents Equal

Since the bases are now the same, set the exponents equal to each other. This gives us the equation \(x = 2\).
03

Verify the Solution

Substitute \(x = 2\) back into the original equation to ensure it is correct: \(8^{2} = 64\). This confirms that \(x = 2\) is the correct solution.

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication of a number by itself. For example, in the expression $$8^2$$, the number 8 is called the base, and the 2 is the exponent. This means that we multiply 8 by itself, which gives us:
  • 8 脳 8 = 64

Exponents are a key concept in understanding various higher-level math topics. When solving equations that involve exponents, it's important to be comfortable with how they work and how to manipulate them. Always keep in mind that exponents tell you how many times to use the base in a multiplication.
Same Base
When solving exponential equations, it's useful to express both sides with the same base. For the problem $$8^x = 64$$, recognizing that 64 can be written as a power of 8 simplifies the process.

Here鈥檚 a step-by-step breakdown:
  • First, identify a common base for both sides. In this instance, 8 is a good choice because 64 can be written as 8 raised to a power: $$64 = 8^2$$
  • Rewrite the original equation with this base: $$8^x = 8^2$$
  • This simplifies the equation as you can now directly compare the exponents.

Making the bases the same allows you to set the exponents equal to each other, greatly simplifying the solution process.
Verification of Solutions
After solving the equation, it鈥檚 crucial to verify your solution to ensure it is correct. In our example, after setting the exponents equal, we found that $$x = 2$$.

To verify:
  • Substitute $$x = 2$$ back into the original equation: $$8^2 = 64$$
  • Calculate: $$8^2 = 8 \times 8 = 64$$
  • Since both sides of the equation match, the solution is confirmed to be correct.

Verification helps catch any potential mistakes and reinforces your understanding of the concepts applied. Double-checking your work is always good practice in math.

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Most popular questions from this chapter

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

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=\sqrt[3]{x+2}\)

Use your calculator to find approximations of the following logarithms. (a) \(\log 356.8\) (b) \(\log 35.68\) (c) \(\log 3.568\) (d) Observe your answers and make a conjecture concerning the decimal values of the common logarithms of numbers greater than 1 that have the same digits.

Graph each exponential function. $$ g(x)=\left(\frac{1}{5}\right)^{x} $$

Let \(k\) represent the number of letters in your last name. (a) Use your calculator to find \(\log k\) (b) Raise 10 to the power indicated by the number in part (a). What is your result? (c) Use the concepts of Section 12.1 to explain why you obtained the answer you found in part (b). Would it matter what number you used for \(k\) to observe the same result?
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