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Magazine Chapter 11: Problem 44
Solve each equation. See Example \(6 .\) $$ \log _{4}(2 x-1)=3 $$
Short Answer
Expert verified
The solution is \( x = 32.5 \).
Step by step solution
01
Use the Definition of a Logarithm
Recall that the logarithm equation \( \log_b(a) = c \) can be rewritten in exponential form as \( b^c = a \). Here, \( b = 4 \), \( a = 2x-1 \), and \( c = 3 \). Thus, rewrite the equation as \( 4^3 = 2x - 1 \).
02
Calculate the Power
Compute \( 4^3 \), which means 4 raised to the power of 3. \( 4^3 = 4 \times 4 \times 4 = 64 \). Hence, the equation becomes \( 64 = 2x - 1 \).
03
Solve for x
Start by isolating \( 2x \) on one side of the equation. Add 1 to both sides to get \( 64 + 1 = 2x \). Simplifying the left side gives \( 65 = 2x \).
04
Divide to Find x
To solve for \( x \), divide both sides of the equation by 2. Thus, \( x = \frac{65}{2} = 32.5 \).
05
Verify the Solution
Substitute \( x = 32.5 \) back into the original equation to check if it holds true. Verify by calculating \( 2x - 1 = 2(32.5) - 1 = 65 - 1 = 64 \), and check \( \log_4(64) \). Since \( 4^3 = 64 \), the solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithms
Logarithms are the inverse operation of exponentiation, serving as a tool to determine the power to which a given base must be raised to produce a certain number. In simple terms, if you have an equation of the form \( \log_b(a) = c \), it means that raising the base \( b \) to the power of \( c \) will yield \( a \).
- This inverse relationship is crucial, enabling us to switch between logarithmic and exponential forms easily.
- For instance, if \( \log_4(64) = 3 \), the equivalent exponential form is \( 4^3 = 64 \).
Exponential Equations
Exponential equations typically involve variables in the exponent, making them a bit more complex to handle than linear equations. They are seen in the form \( b^c = a \), where \( b \) is the base and \( c \) is the exponent; such equations are pivotal in modeling growth and decay processes, among others.
- To solve exponential equations like the one in our exercise, you can often utilize logarithms to bring the variable down from the exponent.
- When manipulating these equations, it is crucial to calculate the exponential power carefully to ensure accuracy, like computing \( 4^3 \) in the given problem, which results in 64.
Problem Solving Techniques
Tackling math problems, especially involving logarithms and exponents, can be simplified using structured problem-solving techniques. These techniques help to systematically break down the problem.
- Step-by-Step Approach: Start by understanding what is being asked. For instance, in logarithmic problems, convert the equation into an exponential form to glean more intuitive insights.
- Calculation and Isolation: Perform necessary calculations like finding exponential values and isolate the variable of interest to determine its value, as shown in the example where \( x = \frac{65}{2} \).
- Verification: Confirm your solution by substituting back into the original equation. This crucial verification step ensures the solution satisfies all initial conditions.
