Math Problem Statement
Use the change-of-base theorem to rewrite the given expression in terms of natural logarithms or common logarithms.
log Subscript 1 divided by 6 Baseline 7log1/6 7equals
Solution
The change-of-base theorem states that for any logarithmic base , and positive numbers and (where ), the logarithm can be rewritten as:
Here, we are tasked to rewrite in terms of natural logarithms () or common logarithms ().
Step-by-Step Solution:
Using the change-of-base formula:
or equivalently:
Now, let’s further simplify using the property of logarithms:
So, substituting back:
Finally:
Final Answer:
Would you like more details or have any questions?
5 Related Questions:
- How does the change-of-base theorem work for other bases like base 10?
- Can you apply the change-of-base theorem to rewrite ?
- Why is rewritten as ?
- How does the change-of-base formula apply to expressions with exponents inside the logarithm?
- What are the advantages of using natural logarithms over other logarithmic bases?
Tip:
Always remember that logarithmic properties, such as , can simplify expressions involving logarithms significantly.
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Math Problem Analysis
Mathematical Concepts
Logarithms
Change-of-Base Formula
Properties of Logarithms
Formulas
log_b a = log_c a / log_c b
ln(a^b) = b * ln(a)
log(1/a) = -log(a)
Theorems
Change-of-Base Theorem
Suitable Grade Level
Grades 9-12