Let's go through each part of the questions step by step.
### Part A: Solve for [tex]\( x \)[/tex] in [tex]\( 5^x = 20 \)[/tex]
To solve for [tex]\( x \)[/tex] in the equation [tex]\( 5^x = 20 \)[/tex], we can use logarithms. Taking the logarithm of both sides (commonly natural logarithm [tex]\( \ln \)[/tex] or common logarithm [tex]\( \log \)[/tex]):
[tex]\[ \log(5^x) = \log(20) \][/tex]
Using the property of logarithms that [tex]\( \log(a^b) = b \log(a) \)[/tex]:
[tex]\[ x \log(5) = \log(20) \][/tex]
Now, solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{\log(20)}{\log(5)} \][/tex]
### Part B: Solve for [tex]\( x \)[/tex] given [tex]\( \log_6 10 = x \)[/tex]
This is a straightforward logarithmic equation given in terms of a logarithm with base 6. To rewrite it:
[tex]\[ x = \log_6 10 \][/tex]
This can be interpreted directly without additional steps.
### Part C: Identify the error in the statement [tex]\( \log_3 8 = x \rightarrow 3^8 = x \)[/tex]
Let's first rewrite the given logarithmic equation correctly:
[tex]\[ \log_3 8 = x \][/tex]
This means that:
[tex]\[ 3^x = 8 \][/tex]
The error in the statement is in the misinterpretation of the logarithmic form. The original statement incorrectly asserts that [tex]\( 3^8 = x \)[/tex], whereas it should be:
[tex]\[ 3^x = 8 \][/tex]
### Part D: Given [tex]\( \log_4 7 \)[/tex]
Rewrite this logarithm in a more familiar base, such as natural log or common log, if desired. This doesn't require further solving since it's just an expression with a given value:
[tex]\[ x = \log_4 7 \][/tex]
### Part E: Calculate [tex]\( \log_4 10 + \log_4 2 \)[/tex]
Using the properties of logarithms, specifically the product rule [tex]\( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)[/tex]:
[tex]\[ \log_4 10 + \log_4 2 = \log_4 (10 \cdot 2) = \log_4 20 \][/tex]
Simplify this equation:
The expression [tex]\( \log_4 10 + \log_4 2 \)[/tex] simplifies correctly to [tex]\( \log_4 20 \)[/tex]. The statement [tex]\( \log_4 10 + \log_4 2 = \log_4 5 \)[/tex] is incorrect.
### Part F: Common Log vs. Natural Log
Common logarithms are base 10, and are denoted as [tex]\( \log \)[/tex] without a base, while natural logarithms are base [tex]\( e \)[/tex] and are denoted as [tex]\( \ln \)[/tex].
For instance:
[tex]\[ \log(100) \approx 2 \][/tex]
[tex]\[ \ln(e^2) = 2 \][/tex]
Both types of logs are useful in different contexts, with common logs being useful in general calculations and natural logs frequently appearing in calculus and natural phenomena applications.
