Textbook Question
Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4.5^2x + 3(5^x) = 28
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6. Exponential & Logarithmic Functions
Solving Exponential and Logarithmic Equations
3:08 minutesProblem 43
Textbook Question
Solve each equation. Give solutions in exact form. See Examples 5–9. ln 4x = 1.5
Verified step by step guidance1
Start by understanding that the equation \( \ln(4x) = 1.5 \) involves a natural logarithm, which is the logarithm to the base \( e \).
To solve for \( x \), you need to eliminate the natural logarithm by exponentiating both sides of the equation. This means you will use the property that if \( \ln(a) = b \), then \( a = e^b \).
Apply this property to the equation: \( 4x = e^{1.5} \).
Now, solve for \( x \) by dividing both sides of the equation by 4: \( x = \frac{e^{1.5}}{4} \).
The solution \( x = \frac{e^{1.5}}{4} \) is in exact form, as required.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Natural Logarithm
The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay. Understanding how to manipulate natural logarithms is essential for solving equations like 'ln(4x) = 1.5'.
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Exponential Equations
Exponential equations are equations in which variables appear as exponents. To solve these equations, one often uses logarithms to isolate the variable. In the context of the given question, converting the logarithmic equation back to its exponential form is a key step in finding the solution.
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Exact Solutions
Exact solutions refer to the precise values of variables without approximation. In the context of logarithmic and exponential equations, this means expressing the solution in terms of logarithms or fractions rather than decimal approximations. This is important for maintaining accuracy in mathematical solutions.
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