Questions: If log81 x = 1/2, find the value of x without a calculator. x =

Transcript text: If $\log _{81} x=\frac{1}{2}$, find the value of $x$ without a calculator. $x=$ $\square$

Solution

Solution Steps

To solve the equation $\log_{81} x = \frac{1}{2}$, we need to convert the logarithmic equation to its exponential form. The exponential form of $\log_b a = c$ is $a = b^c$. Therefore, we can rewrite the given equation as $x = 81^{\frac{1}{2}}$. Since $81$ is $3^4$, we can simplify $81^{\frac{1}{2}}$ to $(3^4)^{\frac{1}{2}}$, which simplifies further to $3^2$. Thus, $x = 9$.

Step 1: Convert Logarithmic Equation to Exponential Form

We start with the equation given by the logarithm: \[ \log_{81} x = \frac{1}{2} \] To convert this logarithmic equation to its exponential form, we rewrite it as: \[ x = 81^{\frac{1}{2}} \]

Step 2: Simplify the Exponential Expression

Next, we simplify $81^{\frac{1}{2}}$. Since $81$ can be expressed as $3^4$, we have: \[ x = (3^4)^{\frac{1}{2}} \] Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we simplify further: \[ x = 3^{4 \cdot \frac{1}{2}} = 3^2 \]

Step 3: Calculate the Final Value

Now, we calculate $3^2$: \[ x = 9 \]