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

Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

문제 97

5장, 문제 97

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

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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