Math Problem Statement
Solution
The question you are asking about is part (c) in the image:
Step-by-Step Solution:
We are given the following information:
- and
We are tasked with finding the value of .
1. Express in terms of logarithms:
Using the logarithmic property , we can break down as: So, .
2. Express in terms of logarithms:
Since , we use the power rule :
3. Substitute into the expression for :
Now, substitute the expressions for and into :
4. Simplify the expression:
Expand the terms: Now subtract : The 's cancel out, leaving:
Final Answer:
This is the simplified value of .
Would you like any further clarifications?
Here are 5 related questions:
- How can the properties of logarithms simplify expressions with products and powers?
- How does the cancellation of terms work when simplifying logarithms?
- How do logarithms help in solving exponential equations?
- What happens when we change the base of logarithms in a problem like this?
- Can the solution method be extended to different logarithmic expressions involving more terms?
Tip: When working with logarithmic equations, always try to break down the logarithms using properties like the product, quotient, and power rules to simplify the terms.
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Math Problem Analysis
Mathematical Concepts
Logarithms
Algebra
Logarithmic Properties
Formulas
log(ab) = log a + log b
log a^n = n log a
Theorems
Logarithmic Product Rule
Logarithmic Power Rule
Suitable Grade Level
Grades 9-12