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Solve the exponential equation 3^x = 81 by expressing each side as a power of the same base and then equating exponents.

The main objective of this question is to solve the exponential equation.

This question uses the concept of exponential equation. Powers can simply be expressed in concise form using exponential expressions. The exponent shows how frequently the base is utilized as a factor.

Expert Answer

We are given:

\[\space 3^x \space = \space 81 \]

We can also write it as:

\[\space 81 \space = 9 \space \times \space 9 \]

\[\space = \space 3 \space \times \space 3 \times \space 3 \space \times \space 3 \]

Then:

\[\space 81 \space = \space 3^4 \]

Now:

\[^\space 3^x \space = \space 3^4 \]

We know that:

\[\space a^m \space = \space a^n \space , \space a\neq 0 \]

Then:

\[\space x \space = \space 4 \]

The final answer is:

\[\space 3^x \space = \space 81 \]

Where $ x $ is equal to $ 4$ .

Numerical Results

The value of $ x $ in the given exponential equation is $ 3 $ .

Example

Find the value of $ x $ in the given exponential expressions.

  • \[\space 3^x \space = \space 2 4 3 \]
  • \[\space 3^x \space = \space 7 2 9 \]
  • \[\space 3^x \space = \space 2 1 8 7 \]

We are given that:

\[\space 3^x \space = \space 2 4 3 \]

We can also write as:

\[\space 2 4 3 \space = 9 \space \times \space 9 \space \times \space 3 \]

\[\space = \space 3 \space \times \space 3 \times \space 3 \space \times \space 3  \space \times \space 3 \]

Then:

\[\space 2 4 3 \space = \space 3^5 \]

Now:

\[\space 3^x \space = \space 3^5 \]

We know that:

\[\space a^m \space = \space a^n \space , \space a \neq 0 \]

Then:

\[\space x \space = \space 5 \]

The final answer is:

\[\space 3^x \space = \space 2 4 3 \]

Where $ x $ is equal to $ 5$ .

Now we have to solve it for the second exponential equation.

We are given that:

\[\space 3^x \space = \space 7 2 9 \]

We can also write as:

\[\space = \space 3 \space \times \space 3 \times \space 3 \space \times \space 3  \space \times \space 3 \space \times \space 3    \]

Then:

\[\space 7 2 9 \space = \space 3^6 \]

Now:

\[^\space 3^x \space = \space 3^6 \]

We know that:

\[\space a^m \space = \space a^n \space , \space a \neq 0 \]

Then:

\[\space x \space = \space 6 \]

The final answer is:

\[\space 3^x \space = \space 7 2 9 \]

Where $ x $ is equal to $ 6$ .

Now we have to solve it for the third expression.

We are given that:

\[\space 3^x \space = \space 2 1 8 7 \]

We can also write as:

\[\space = \space 3 \space \times \space 3 \times \space 3 \space \times \space 3  \space \times \space 3 \space \times \space 3 \space \times \space 3    \]

Then:

\[\space 2 1 8 7\space = \space 3^7 \]

Now:

\[\space 3^x \space = \space 3^7 \]

We know that:

\[\space a^m \space = \space a^n \space , \space a \neq 0 \]

Then:

\[\space x \space = \space 7 \]

The final answer is:

\[\space 3^x \space = \space 2 1 8 7 \]

where $ x $ is equal to $ 7 $ .

 

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