Math Problem Statement
Solution
Let's break down and solve both questions one by one.
Question (22):
The problem states:
We need to find in terms of and .
Step 1: Prime Factorization of 12 We can express as:
So, we have:
By using the logarithmic property:
We get:
Step 2: Change of Base Formula We need to express and in terms of base 10 (since we are given logs in base 10). Using the change of base formula:
We can rewrite the logarithms:
So:
Factor out :
This matches option B, which is under the assumption .
Question (23):
The problem asks to compute:
Step 1: Simplify the Logarithms We know that:
And:
Step 2: Perform the Division Now, divide the two results:
This matches option A, , not B.
Would you like more details on any of these solutions?
Here are 5 related questions to help deepen your understanding:
- How does the change of base formula work for logarithms?
- Can you explain the logarithmic properties used for simplifying products and powers?
- How would you solve using the same change of base method?
- What happens if you use natural logarithms instead of base 10?
- How can you express in terms of and ?
Tip: Always remember the properties of logarithms, especially when dealing with different bases—change of base and product-to-sum rules are extremely useful!
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Math Problem Analysis
Mathematical Concepts
Logarithms
Change of Base Formula
Logarithmic Identities
Formulas
log_b(xy) = log_b(x) + log_b(y)
log_b(x^n) = n*log_b(x)
Change of Base: log_b(x) = log_c(x)/log_c(b)
Theorems
Properties of Logarithms
Change of Base Formula
Suitable Grade Level
Grades 9-10