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Chapter 12: Problem 97

Solve. $$ \log _{3} x=2 $$

Short Answer

Expert verified
x = 9

Step by step solution

01

- Understand the Logarithmic Form

The given equation is in logarithmic form: \(\log _{3} x = 2\). This means that the logarithm of x to the base 3 is equal to 2.
02

- Convert Logarithmic Form to Exponential Form

To solve the equation, we need to convert the logarithmic form to the exponential form. The equation \(\log _{b} a = c\) can be rewritten as \(a = b^c\). Applying this to our equation: \(\log _{3} x = 2\) becomes \(x = 3^2\).
03

- Solve the Exponential Equation

Solve for x by calculating \(3^2\): \(x = 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Form
The equation \( \( \log _{3} x = 2 \)\) is in logarithmic form. This tells us that the logarithm of \( x \) to the base 3 equals 2. Understanding the logarithmic form is key in recognizing how logarithms relate to exponential functions. Logarithms essentially ask the question: 'To what power must the base be raised, to obtain the given number?' In this equation, the logarithmic form shows us that 3 raised to some power results in \( x \). Here, \( \log_{b} a = c \) can be interpreted as the power (c) needed to raise the base (b) to get the number (a).
Exponential Form
To make solving easier, let's convert the logarithmic equation to exponential form. Recall the relationship: \( \log_{b} a = c \implies a = b^c \). Applying this to our problem, \( \log_{3} x = 2 \) becomes \( x = 3^2 \). The base (3) raised to the power (2) results in \( x \). Converting to exponential form helps to straightforwardly express the relationship between the base, exponent, and the number. This step turns the logarithmic question into a simpler multiplication question.
Solving the Equation
Now we solve the exponential equation derived in the previous step: \( x = 3^2 \). By calculating, \( 3^2 = 9 \). Therefore, \( x = 9 \). This final step involves basic arithmetic, ensuring you arrive at the correct solution. Solving logarithmic equations often starts with understanding the logarithmic form, converting it to exponential form, and then performing basic computations. Mastering this process will help you tackle similar logarithmic and exponential equations effectively.

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Most popular questions from this chapter

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