Solve each equation. log₉ x = 5/2

Verified step by step guidance

1. Recognize that the equation is given in logarithmic form: \(\log_{9} x = \frac{5}{2}\). This means the logarithm base is 9, and the logarithm of \(x\) equals \(\frac{5}{2}\).
2. Recall the definition of logarithm: \(\log_{a} b = c\) means \(a^{c} = b\). Apply this to rewrite the equation as an exponential equation: \(9^{\frac{5}{2}} = x\).
3. Express the base 9 as a power of a smaller base if possible. Since \$9 = 3^{2}$, rewrite \(9^{\frac{5}{2}}\) as \((3^{2})^{\frac{5}{2}}\).
4. Use the power of a power property: \((a^{m})^{n} = a^{m \times n}\). So, \((3^{2})^{\frac{5}{2}} = 3^{2 \times \frac{5}{2}}\).
5. Simplify the exponent multiplication: \(2 \times \frac{5}{2} = 5\). Therefore, \(x = 3^{5}\). This expresses the solution for \(x\) in exponential form.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: to what power must the base be raised to produce a given number? For example, log₉(x) = y means 9 raised to y equals x.

Properties of Logarithms

Understanding properties like the definition of logarithms and how to rewrite logarithmic equations in exponential form is essential. For instance, logₐ(b) = c can be rewritten as a^c = b, which helps in solving equations.

Solving Exponential Equations

Once the logarithmic equation is converted to exponential form, solving involves applying exponent rules and algebraic manipulation to isolate the variable. This step is crucial to find the exact value of x.