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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 97
Chapter 5, Problem 97

Solve each equation. 3x2=453^{x^2} = 45

Verified step by step guidance
1
Identify the equation given: \$3^{x^2} = 45\(. Our goal is to solve for \)x$.
Take the natural logarithm (or log base 10) of both sides to help bring down the exponent. This gives: \(\ln(3^{x^2}) = \ln(45)\).
Use the logarithmic property that allows you to move the exponent in front: \(x^2 \cdot \ln(3) = \ln(45)\).
Isolate \(x^2\) by dividing both sides by \(\ln(3)\): \(x^2 = \frac{\ln(45)}{\ln(3)}\).
Finally, solve for \(x\) by taking the square root of both sides: \(x = \pm \sqrt{\frac{\ln(45)}{\ln(3)}}\). Remember to consider both the positive and negative roots.

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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, such as 3^(x^2) = 45. Solving these requires understanding how to manipulate and isolate the exponential expression to find the variable's value.
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Logarithms

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms allows you to rewrite the equation in a form that makes the exponent accessible for solving.
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Solving Quadratic Equations

When the exponent is a quadratic expression like x^2, after applying logarithms, you often get a quadratic equation. Solving this requires techniques such as factoring, completing the square, or using the quadratic formula to find the values of x.
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