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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 39
Chapter 5, Problem 39

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. 52x + 3(5x) = 28

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1
Recognize that the equation involves exponential expressions with the same base, 5. The equation is \(5^{2x} + 3(5^x) = 28\).
Rewrite \$5^{2x}\( as \)(5^x)^2\( to express the equation in terms of a single variable. Let \)y = 5^x\(, so the equation becomes \)y^2 + 3y = 28$.
Rewrite the equation as a quadratic equation by moving all terms to one side: \(y^2 + 3y - 28 = 0\).
Solve the quadratic equation \(y^2 + 3y - 28 = 0\) using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=3\), and \(c=-28\).
After finding the values of \(y\), substitute back \(y = 5^x\) and solve for \(x\) by taking the logarithm base 5: \(x = \log_5(y)\). Calculate the decimal values of \(x\) to the nearest thousandth.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

Exponential equations involve variables in the exponent position, such as 5^x. Solving these requires understanding how to manipulate and rewrite expressions to isolate the variable, often by expressing terms with the same base or using substitution.
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Substitution Method

When an equation contains terms like 5^(2x) and 5^x, substitution simplifies the problem. For example, letting y = 5^x transforms the equation into a quadratic form, making it easier to solve using algebraic methods.
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Rounding Irrational Solutions

Some solutions to exponential equations are irrational numbers. These should be approximated as decimals rounded to a specified place value, such as the nearest thousandth, to provide a practical and understandable answer.
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