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Chapter 4: Problem 77

Solve each equation. $$10^{x-1}=0.01$$

Short Answer

Expert verified
x = -1

Step by step solution

01

Express 0.01 as a Power of 10

Recognize that 0.01 can be written as a power of 10. In this case, \( 0.01 = 10^{-2} \).
02

Set the Exponents Equal

Since the bases are the same (both are 10), set the exponents equal to each other. This gives \( x - 1 = -2 \).
03

Solve for x

Solve the equation for x by adding 1 to both sides: \( x - 1 + 1 = -2 + 1 \). Simplifying this gives \( x = -1 \).

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Key Concepts

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

Exponential Notation
Exponential notation is a shorthand way to express very large or very small numbers by using powers of a given base.
It's especially useful when dealing with powers of ten. In exponential notation, the number is written as a base raised to an exponent.
For example, instead of writing 1000, you can write it as \(10^3\). Here, 10 is the base and 3 is the exponent, indicating that the base is multiplied by itself three times: \(10 \times 10 \times 10\).
This notation simplifies calculations and comparisons, particularly in scientific and mathematical contexts.
Powers of Ten
Powers of ten are a specific application of exponential notation where the base is 10.
Understanding powers of ten helps in interpreting scientific notation and dealing with very large or small numbers.
When you see a number like \(10^2\), it means you multiply 10 by itself twice: 10 times 10, which equals 100. Conversely, \(10^{-2}\) means you divide 1 by 10 twice: \(\frac{1}{10} \times \frac{1}{10} = 0.01\).
Recognizing these patterns can make many calculations simpler and faster.
  • \(10^0 = 1\)
  • \(10^1 = 10\)
  • \(10^2 = 100\)
  • \(10^{-1} = 0.1\)

Powers of ten are frequently used in scientific notation to express values compactly and clearly. For example, 3000 can be written as \(3 \times 10^3\).
Equating Exponents
Equating exponents is a key step in solving exponential equations.
When two expressions with the same base are equal, their exponents must be equal as well. This principle can be used to solve for unknown variables.
Consider the equation \(10^{x-1} = 0.01\). First, recognize that 0.01 is the same as \(10^{-2}\). So, we rewrite the equation as \(10^{x-1} = 10^{-2}\).
Since the base (10) is the same on both sides, we can set the exponents equal to each other: \(x-1 = -2\).
By solving this simplified equation, we find that \(x = -1\).
Following these methods ensures that you properly isolate and solve for the variable in any exponential equation.

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Most popular questions from this chapter

Solve each equation. Find the exact solutions. $$\log _{x}(9)=\frac{1}{2}$$

Solve each problem. Find the annual percentage rate compounded continuously to the nearest tenth of a percent for which \(\$ 10\) would grow to \(\$ 30\) for each of the following time periods. a. 5 years b. 10 years c. 20 years d. 40 years

Solve each equation. Find the exact solutions. $$\log _{2}\left(\log _{3}\left(\log _{4}(x)\right)\right)=0$$

Solve each problem. Ben Franklin's gift of \(\$ 4000\) to the city of Philadelphia in 1790 was not managed as well as his gift to Boston. The Philadelphia fund grew to only \(\$ 2\) million in 200 years. Find the annual rate compounded continuously that would yield this total value.

Find the approximate solution to each equation. Round to four decimal places. $$10^{3 x}=5$$
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