To solve the equation [tex]\(\log_8 12 = x - 2\)[/tex], follow these steps:
1. Rewrite the equation in its exponential form: Recall that the logarithmic equation [tex]\(\log_a b = c\)[/tex] is equivalent to the exponential equation [tex]\(a^c = b\)[/tex]. Here, [tex]\(a\)[/tex] is the base of the logarithm, [tex]\(b\)[/tex] is the value, and [tex]\(c\)[/tex] is the exponent.
In this instance, we have:
[tex]\[\log_8 12 = x - 2\][/tex]
2. Convert to exponential form:
Since [tex]\(\log_8 12 = x - 2\)[/tex]:
[tex]\[8^{(x - 2)} = 12\][/tex]
3. Isolate [tex]\(x\)[/tex]: To find [tex]\(x\)[/tex], we first need to solve for [tex]\(x - 2\)[/tex]. We know we can take the logarithm of both sides to simplify the problem. Using the change of base formula for logarithms, we get:
[tex]\[x - 2 = \log_8 12\][/tex]
4. Apply the change of base formula: The change of base formula for logarithms states [tex]\(\log_a b = \frac{\log_c b}{\log_c a}\)[/tex], where [tex]\(c\)[/tex] is any positive number (commonly 10 or [tex]\(e\)[/tex] for natural logarithms). Here, we use the natural logarithm (base [tex]\(e\)[/tex]):
[tex]\[x - 2 = \frac{\log 12}{\log 8}\][/tex]
5. Calculate the value: Use a calculator to find the values of [tex]\(\log 12\)[/tex] and [tex]\(\log 8\)[/tex]:
[tex]\[\log 12 \approx 2.4849\][/tex]
[tex]\[\log 8 \approx 2.0794\][/tex]
Thus,
[tex]\[x - 2 = \frac{2.4849}{2.0794} \approx 1.1949875002403856\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[x = 1.1949875002403856 + 2 \approx 3.1949875002403854\][/tex]
7. Round the answer: Round the answer to four decimal places:
[tex]\[x \approx 3.1950\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies [tex]\(\log_8 12 = x - 2\)[/tex], rounded to four decimal places, is:
[tex]\[x \approx 3.1950\][/tex]
