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Chapter 9: Problem 44

Solve each equation. Check your solutions. \(\log _{9} x=2\)

Short Answer

Expert verified
x = 81; Solution verified.

Step by step solution

01

Understand the Equation

We are given the equation \(\log_{9} x = 2\). This is a logarithmic equation where the base of the logarithm is 9, and we need to find the value of \(x\).
02

Convert to Exponential Form

To solve the logarithmic equation, convert it to its exponential form. Recall that \(\log_{b} a = c\) is equivalent to \(b^{c} = a\). Applying this to our equation, we have \(9^{2} = x\).
03

Calculate the Exponential Value

Compute \(9^{2}\) to find the value of \(x\). We know that \(9^{2} = 9 \times 9 = 81\). So, \(x = 81\).
04

Check the Solution

Substitute \(x = 81\) back into the original equation to verify the solution. Calculate \(\log_{9} 81\) and verify if it equals 2. Since \(9^{2} = 81\), indeed, \(\log_{9} 81 = 2\), confirming that our solution is correct.

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Key Concepts

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

Exponential Form
When dealing with logarithmic equations, one powerful tool is conversion to exponential form. This helps to simplify and solve the equation. The general rule is if you have
  • \(\log_{b} a = c\)
then you can convert it to its exponential form
  • \(b^{c} = a\)
In our example, we have \(\log_{9} x = 2\). Using the rule, convert it to the exponential form \(9^{2} = x\). Thus, we find that \(x = 81\). This conversion is key to unlocking the solution as it changes the problem from a logarithmic form to a simple exponentiation problem.
Logarithmic Function
The logarithmic function is the inverse of the exponential function. A logarithm answer the question: "To what exponent must the base be raised, to yield a certain number?"
For example, \(\log_{9} x = 2\) asks: "To what power must 9 be raised, to get \(x\)?"
  • Logarithms are denoted as \(\log_{b} a\), where \(b\) is the base, \(a\) is the result of raising \(b\) to some power, and the logarithm gives that power.
  • In our case, \(b = 9\), \(a = x\), and the power is 2.
The logarithmic function is very useful in solving problems involving multiplicative growth or decay, such as in compound interest, population growth, and more. By understanding logarithms, you can reverse engineer exponential growth or determine the time it takes for something to grow a specific amount.
Checking Solutions
After solving an equation, it's crucial to verify if the solution is correct. Checking solutions ensures that no errors have been made in the process.
In our example, after calculating \(x = 81\), we substitute it back into the original equation \(\log_{9} x = 2\) to check:
  • Calculate \(\log_{9} 81\) and verify if it equals 2.
  • Since \(9^{2} = 81\), the condition holds true, confirming \(\log_{9} 81 = 2\).
Checking solutions is a vital step in any mathematical process as it provides final confirmation and builds confidence that you have accurately understood and solved the problem at hand.

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