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Chapter 10: Problem 47

Solve each equation. $$\log _{x} 9=\frac{1}{2}$$

Short Answer

Expert verified
x = 81

Step by step solution

01

Understanding the Logarithmic Equation

The given equation is \(\log_{x} 9 = \frac{1}{2}\). This means that \(x\) raised to the power of \(\frac{1}{2}\) equals \(9\).
02

Convert the Logarithmic Equation to Exponential Form

Rewrite the logarithmic equation in its exponential form: \[ x^{\frac{1}{2}} = 9 \]
03

Solve for x by Squaring Both Sides

To solve for \(x\), square both sides of the equation to eliminate the fractional exponent: \[ \left(x^{\frac{1}{2}}\right)^{2} = 9^{2} \] Simplify this to get: \[ x = 81 \]

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Key Concepts

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

Logarithms
A logarithm is a way to express the power to which a number (the base) must be raised to obtain another number. For example, in the equation \( \log_{x} 9 = \frac{1}{2}\), the base is \(x\), and 9 is the number we want to obtain by raising the base to the power \( \frac{1}{2}\). Logarithms are particularly useful for solving equations where the variable is in the exponent. The general form of a logarithmic equation is:
\[ \log_{b}(a) = c \] Here, \(b\) is the base, \(a\) is the result, and \(c\) is the exponent.
Exponential Form
Converting a logarithmic equation to its exponential form is a crucial step in solving logarithmic equations. The exponential form of a logarithmic equation \(\log_{b}(a) = c\) is \[ b^{c} = a \]
This means that if you raise the base \(b\) to the power \(c\), you will get the number \(a\). In our example, the logarithmic equation is: \ \(\ \log_{x} 9 = \frac{1}{2}\ \) . To convert this into exponential form, we rewrite it as:
\[ x^{\frac{1}{2}} = 9 \]
This shows that we need to determine the value of \(x\) which, when raised to the power of \( \frac{1}{2}\), equals 9.
Solving Equations
To solve for \(x\) in our converted exponential equation \( \ x^{\frac{1}{2}} = 9\), we proceed by eliminating the fractional exponent. Here's how:
1. **Square both sides** of the equation to get rid of the \( \frac{1}{2}\) exponent:
\[ (x^{\frac{1}{2}})^{2} = 9^{2} \]
2. **Simplify** the equation:
\[ x^{\frac{1}{2}*2} = 81 \]
3. \( x^{1} = 81 \) means \( x = 81\)
Thus, we find that \( x = 81 \). Solving logarithmic equations often requires converting to exponential form and using algebraic techniques to isolate the variable.

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Most popular questions from this chapter

Graph each logarithmic function. $$g(x)=\log _{1 / 4} x$$

How long, to the nearest hundredth of a year, would it take an initial principal \(P\) to double if it were invested at \(2.5 \%\) compounded continuously?

What will be the amount \(A\) in an account with initial principal \(\$ 4000\) if interest is compounded continuously at an annual rate of \(3.5 \%\) for 6 yr?

Solve each problem. Suppose that \(\$ 3000\) is deposited at \(3.5 \%\) compounded quarterly. (a) How much money will be in the account at the end of 7 yr? (Assume no withdrawals are made.) (b) To one decimal place, how long will it take for the account to grow to \(\$ 5000 ?\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{3 x}=9 $$
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