Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.5: Properties Of Logarithms
Problem 73E
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Question
![### Converting Logarithmic Equations to Exponential Form
When converting a logarithmic equation to its exponential form, we use the relationship between logarithms and exponents. Here is an example to illustrate this conversion:
#### Given Logarithmic Equation:
\[ \log 100,000 = 5 \]
#### Step-by-Step Explanation:
1. **Identify the Components:**
- The base of the logarithm is assumed to be 10 (since no base is explicitly mentioned).
- 100,000 is the argument (or the number we're taking the logarithm of).
- 5 is the result of the logarithm.
2. **Apply the Logarithm Relationship:**
- The general form of a logarithmic equation is \(\log_b (a) = c\), which can be rewritten in exponential form as \(b^c = a\).
3. **Rewrite in Exponential Form:**
- For the given equation \(\log 100,000 = 5\), we assume the base (\(b\)) is 10.
- Hence, the equation in exponential form is:
\[ 10^5 = 100,000 \]
### Conclusion:
The equation \(\log 100,000 = 5\) can be rewritten as \(10^5 = 100,000\).
Understanding this conversion is fundamental in solving logarithmic and exponential equations, which are extensively used in various fields such as mathematics, science, and engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15430a79-cfc8-483e-83f7-796f8bb05713%2Fda3d3d52-7d17-4c93-a581-7a7d8e7381ca%2Fuujd8th_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Converting Logarithmic Equations to Exponential Form
When converting a logarithmic equation to its exponential form, we use the relationship between logarithms and exponents. Here is an example to illustrate this conversion:
#### Given Logarithmic Equation:
\[ \log 100,000 = 5 \]
#### Step-by-Step Explanation:
1. **Identify the Components:**
- The base of the logarithm is assumed to be 10 (since no base is explicitly mentioned).
- 100,000 is the argument (or the number we're taking the logarithm of).
- 5 is the result of the logarithm.
2. **Apply the Logarithm Relationship:**
- The general form of a logarithmic equation is \(\log_b (a) = c\), which can be rewritten in exponential form as \(b^c = a\).
3. **Rewrite in Exponential Form:**
- For the given equation \(\log 100,000 = 5\), we assume the base (\(b\)) is 10.
- Hence, the equation in exponential form is:
\[ 10^5 = 100,000 \]
### Conclusion:
The equation \(\log 100,000 = 5\) can be rewritten as \(10^5 = 100,000\).
Understanding this conversion is fundamental in solving logarithmic and exponential equations, which are extensively used in various fields such as mathematics, science, and engineering.
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