Math Problem Statement
Solution
To solve the equation using the change of base formula, we will proceed step-by-step:
Step 1: Isolate the Exponential Expression
Start by isolating the term involving the exponential expression:
Add 4 to both sides:
Step 2: Apply Logarithm to Both Sides
To solve for , take the logarithm of both sides. We can use any logarithm, but for consistency, let's use the natural logarithm (base ):
Step 3: Use the Logarithm Power Rule
Apply the power rule of logarithms, which states :
Step 4: Solve for
Now, solve for by dividing both sides by :
Step 5: Use the Change of Base Formula
The change of base formula for logarithms states that:
In this case, using natural logarithms () or common logarithms () provides the same solution. Therefore:
Step 6: Calculate the Value of
Now, compute the value using a calculator:
Step 7: Round to the Nearest Thousandth
Rounded to the nearest thousandth, the value of is:
Final Answer
The solution for is approximately .
Would you like more details or have any questions?
Related Questions:
- How do you solve exponential equations that involve different bases?
- Can you solve using a logarithm?
- What is the power rule of logarithms, and how is it applied?
- How does the change of base formula work for logarithms other than base 10 or ?
- What are the differences between natural logarithms () and common logarithms ()?
Tip:
When solving equations involving exponents, always consider using logarithms to simplify and isolate the variable.
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Equations
Logarithms
Formulas
Change of Base Formula: log_b(y) = log_k(y) / log_k(b)
Logarithm Power Rule: ln(a^b) = b * ln(a)
Theorems
Change of Base Theorem
Logarithm Power Rule
Suitable Grade Level
Grades 10-12