Domanda del libro di testo
Evaluate or simplify each expression without using a calculator.
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 97
Solve each equation.
Guida verificata passo dopo passo1
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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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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Solving Exponential Equations Using Logs
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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7:30Logarithms Introduction
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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06:08Solving Quadratic Equations by Factoring
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