Math  /  Algebra

QuestionSolve for xx : log3(x5)=9x=\begin{array}{l} \log _{3}\left(x^{5}\right)=9 \\ x=\square \end{array}
You may enter the exact value or round to 4 decimal places. Question Help: Video Message instructor

Studdy Solution

STEP 1

1. The equation involves a logarithmic function with base 3.
2. We will use properties of logarithms and exponents to solve for x x .

STEP 2

1. Convert the logarithmic equation to an exponential equation.
2. Solve the resulting exponential equation for x x .
3. Simplify the expression to find the exact value of x x .

STEP 3

Convert the logarithmic equation to an exponential form. The equation log3(x5)=9\log_{3}(x^{5}) = 9 can be rewritten using the definition of a logarithm:
x5=39 x^{5} = 3^{9}

STEP 4

Solve the exponential equation for x x . To isolate x x , take the fifth root of both sides:
x=(39)15 x = (3^{9})^{\frac{1}{5}}

STEP 5

Simplify the expression. Use the property of exponents (am)n=amn(a^{m})^{n} = a^{m \cdot n}:
x=395 x = 3^{\frac{9}{5}}

STEP 6

Calculate the exact value or round to 4 decimal places. Using a calculator:
x31.814.3485 x \approx 3^{1.8} \approx 14.3485
The value of x x is approximately:
14.3485 \boxed{14.3485}

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