Get started for free
Log In Start studying!
Get started for free Log out
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 \(\log_{10}(1000) = 3\).
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’s 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’s 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’ll find that converting between the forms becomes easier over time.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
Most popular questions from this chapter
Recommended explanations on Math Textbooks
Statistics
Read ExplanationApplied Mathematics
Read ExplanationDiscrete Mathematics
Read ExplanationProbability and Statistics
Read ExplanationPure Maths
Read ExplanationCalculus
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.