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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
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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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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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Logarithms 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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Solving Quadratic Equations by Factoring
Related Practice
Textbook Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log6(x−1x2+4)=log6(x−1)−log6(x2+4)\(\log\)_6 \(\left\)( \(\frac{x - 1}{x^2 + 4}\) \(\right\)) = \(\log\)_6 (x - 1) - \(\log\)_6 (x^2 + 4)

Textbook Question

Evaluate or simplify each expression without using a calculator. eln5x2e^{\(\ln\)5x^2}

Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln x + ln(2x) = ln(3x)

Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. [log(x+2)log(x−1)]=log(x+2)−log(x−1)[\(\frac{log(x+2)}{log(x-1)}\)]=log(x+2)-log(x-1)

Textbook Question

Solve each equation. 3|log x|−6=0

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Textbook Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log(x + 3) - log(2x) = [log(x + 3)/log(2x)]

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