Certainly! Let's solve the exponential equation [tex]\( 23^x = 6 \)[/tex] step by step.
1. Understanding the Equation:
The equation we need to solve is [tex]\( 23^x = 6 \)[/tex]. We need to find the value of [tex]\( x \)[/tex] that makes this equation true.
2. Applying the Logarithm:
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (or common logarithm) of both sides of the equation. This is because taking the logarithm allows us to bring the exponent down as a multiplier, making it easier to solve for [tex]\( x \)[/tex].
[tex]\[
\log(23^x) = \log(6)
\][/tex]
3. Using Logarithm Properties:
By using the power rule of logarithms, which states that [tex]\( \log(a^b) = b \log(a) \)[/tex], we can rewrite the left-hand side of the equation:
[tex]\[
x \cdot \log(23) = \log(6)
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
Now, we need to isolate [tex]\( x \)[/tex]. To do this, we divide both sides of the equation by [tex]\( \log(23) \)[/tex]:
[tex]\[
x = \frac{\log(6)}{\log(23)}
\][/tex]
5. Calculating the Value:
Using a calculator to evaluate the right-hand side, we get:
[tex]\[
x \approx 0.5714440358797147
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 23^x = 6 \)[/tex] is approximately [tex]\( 0.5714440358797147 \)[/tex].
By following these steps, we have determined that the solution to the equation [tex]\( 23^x = 6 \)[/tex] is [tex]\( x \approx 0.5714440358797147 \)[/tex].
