To approximate the value of [tex]\(\log_b 18\)[/tex] using the given values [tex]\(\log_b 6 = 0.921\)[/tex] and [tex]\(\log_b 3 \approx 0.565\)[/tex], we can utilize the properties of logarithms, particularly the property that states the logarithm of a product is the sum of the logarithms of the factors.
Given:
[tex]\[
\log_b 6 = 0.921
\][/tex]
and
[tex]\[
\log_b 3 \approx 0.565
\][/tex]
We need to find [tex]\(\log_b 18\)[/tex]. Notice that [tex]\(18\)[/tex] can be expressed as the product of [tex]\(6\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
18 = 6 \times 3
\][/tex]
Using the product property of logarithms, we can express [tex]\(\log_b 18\)[/tex] as:
[tex]\[
\log_b 18 = \log_b (6 \times 3)
\][/tex]
According to the product rule for logarithms:
[tex]\[
\log_b (6 \times 3) = \log_b 6 + \log_b 3
\][/tex]
Substituting the given values:
[tex]\[
\log_b 18 = \log_b 6 + \log_b 3
= 0.921 + 0.565
\][/tex]
Adding these values together, we obtain:
[tex]\[
0.921 + 0.565 = 1.486
\][/tex]
Therefore, the approximate value of [tex]\(\log_b 18\)[/tex] is:
[tex]\[
\log_b 18 \approx 1.486
\][/tex]
