Sure, solving the exponential equation [tex]\( 23^x = 6 \)[/tex] involves logarithms and the change of base formula. Let’s go through the steps to find the value of [tex]\( x \)[/tex].
1. Understand the Equation:
The given equation is [tex]\( 23^x = 6 \)[/tex]. We need to solve for [tex]\( x \)[/tex].
2. Apply Logarithms to Both Sides:
To handle the exponent [tex]\( x \)[/tex], we take the logarithm of both sides of the equation. You can use any logarithm base, but the common logarithm (base 10) or natural logarithm (base [tex]\( e \)[/tex]) is usually used for convenience. Here, we will use the common logarithm (base 10):
[tex]\[
\log(23^x) = \log(6)
\][/tex]
3. Use the Power Rule of Logarithms:
According to the power rule of logarithms, [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]. Apply this rule to the left-hand side of the equation:
[tex]\[
x \cdot \log(23) = \log(6)
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], divide both sides of the equation by [tex]\(\log(23)\)[/tex]:
[tex]\[
x = \frac{\log(6)}{\log(23)}
\][/tex]
5. Calculate the Logarithms:
Now, compute the values of [tex]\(\log(6)\)[/tex] and [tex]\(\log(23)\)[/tex]:
[tex]\[
\log(6) \approx 0.778151
\][/tex]
[tex]\[
\log(23) \approx 1.361728
\][/tex]
6. Divide the Logarithms:
Finally, divide the logarithm of 6 by the logarithm of 23 to find [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{0.778151}{1.361728} \approx 0.571444
\][/tex]
Therefore, the solution to the equation [tex]\( 23^x = 6 \)[/tex] is approximately [tex]\( x \approx 0.571444 \)[/tex].
