Sure, let's go through the steps to solve the exponential equation [tex]\(23^x = 6\)[/tex] using the change of base formula, as mentioned in the hint.
1. Understand the Problem:
The exponential equation we need to solve is [tex]\(23^x = 6\)[/tex]. Our goal is to find the value of [tex]\(x\)[/tex].
2. Apply the Change of Base Formula:
To solve for [tex]\(x\)[/tex], we can use the change of base formula, which states: [tex]\(\log_b(y) = \frac{\log(y)}{\log(b)}\)[/tex]. In our case, [tex]\(b = 23\)[/tex] and [tex]\(y = 6\)[/tex].
3. Rewrite the Equation:
We can rewrite the equation [tex]\(23^x = 6\)[/tex] in logarithmic form:
[tex]\[
x = \log_{23}(6)
\][/tex]
Using the change of base formula, this can be expressed as:
[tex]\[
x = \frac{\log(6)}{\log(23)}
\][/tex]
4. Calculate the Logarithms:
Find the logarithms of 6 and 23 using natural logarithms (logarithms base [tex]\(e\)[/tex]):
[tex]\[
\log(6) \approx 1.791759469228055
\][/tex]
[tex]\[
\log(23) \approx 3.1354942159291497
\][/tex]
5. Divide the Logarithms:
Now, we divide the logarithm of 6 by the logarithm of 23 to find [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1.791759469228055}{3.1354942159291497} \approx 0.5714440358797147
\][/tex]
6. Conclusion:
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 solved the equation [tex]\(23^x = 6\)[/tex], and the solution is [tex]\(x \approx 0.5714440358797147\)[/tex].
