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Chapter 10: Problem 130

In the following exercises, convert from exponential to logarithmic form. $$ 10^{3}=1000 $$

Short Answer

Expert verified
\(\log_{10}(1000) = 3\)

Step by step solution

01

- Identify the base

In the given exponential form, identify the base of the exponent. Here, the base is 10.
02

- Identify the exponent

Identify the exponent in the given exponential expression. Here, the exponent is 3.
03

- Identify the result

Identify the result of the exponential expression. Here, the result is 1000.
04

- Convert to logarithmic form

To convert the exponential form to logarithmic form, use the general conversion structure: Exponential form: \(b^e = N\), Logarithmic form: \(\log_b(N) = e\). Thus, for our example, \(10^3 = 1000\) becomes \(\log_{10}(1000) = 3\).

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Key Concepts

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

Exponential Expressions
Let's start with understanding exponential expressions. In an exponential expression, you have a number that is raised to the power of another number. For example, in the expression \(10^{3} = 1000\), 10 is called the base and 3 is the exponent. The value 1000 is the result of raising 10 to the power of 3.
Here are some more examples to help you grasp the concept better:
  • \(2^{5} = 32\)
  • \(3^{4} = 81\)
  • \(5^{2} = 25\)
An exponential expression helps to quickly show repeated multiplication by the same number. Instead of writing \(10 \times 10 \times 10 = 1000\), you simply write \(10^{3} = 1000\).
Logarithmic Forms
Now let鈥檚 dive into logarithmic forms. A logarithm essentially does the reverse operation of an exponent. It tells us what exponent we need to raise the base to in order to get a certain number. For instance, in the expression \(\text{log}_{10}(1000) = 3\), 10 is the base, 1000 is the result, and 3 is the exponent.
This means you have to raise 10 to the power of 3 to get 1000. Here are more examples:
  • \(\text{log}_{2}(32) = 5 \) because \(2^{5} = 32\)
  • \(\text{log}_{3}(81) = 4 \) because \(3^{4} = 81\)
  • \(\text{log}_{5}(25) = 2 \) because \(5^{2} = 25\)
Understanding this concept is key to converting between exponential and logarithmic forms.
Bases and Exponents
To fully grasp exponential and logarithmic forms, you must understand the roles of bases and exponents. In the expression \(b^{e} = N\):
  • The base (b) is the number that you repeatedly multiply.
  • The exponent (e) tells you how many times to multiply the base by itself.
  • The result (N) is the outcome of the multiplication.
For example:
  • In \(2^{3} = 8\), 2 is the base, 3 is the exponent, and 8 is the result.
  • In \(5^{4} = 625\), 5 is the base, 4 is the exponent, and 625 is the result.
When converting to logarithmic form, you simply rearrange these elements:
The equation \(b^{e} = N\) becomes \(\text{log}_{b}(N) = e\).
Mathematical Conversions
Converting between exponential and logarithmic forms is straightforward once you understand the basic structures. Here鈥檚 a step-by-step guide using our original exercise for better clarity:
  • Step 1: Start with the exponential form \(10^{3} = 1000\)
  • Step 2: Identify the base (10), exponent (3), and result (1000)
  • Step 3: Use the conversion structure: \(\text{log}_{b}(N) = e\)
  • Step 4: Apply it to your example to get \(\text{log}_{10}(1000) = 3\)
This conversion follows the pattern \(b^{e} = N\) to \(\text{log}_{b}(N) = e\). By practicing this technique with different numbers, you鈥檒l find that converting between the forms becomes easier over time.

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Most popular questions from this chapter

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(4 \log _{2} x+6 \log _{2} y\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log _{4} \sqrt{x}\)

In the following exercises, solve each equation. \(\log _{5}(3 x-8)=2\)

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log 4+\log 25\)
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