To find the approximate value of [tex]\(\log_3 48\)[/tex] using the given values [tex]\(\log 48 \approx 1.68\)[/tex] and [tex]\(\log 3 \approx 0.48\)[/tex], we can use the change of base formula. This formula is helpful when we want to convert a logarithm of one base to another.
The change of base formula states:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are positive real numbers and [tex]\(c\)[/tex] is the new base for the logarithms, typically 10 (common logarithm) unless specified otherwise.
In this problem, we are asked to find [tex]\(\log_3 48\)[/tex]. Using the change of base formula, we get:
[tex]\[
\log_3 48 = \frac{\log_{10} 48}{\log_{10} 3}
\][/tex]
We are given:
[tex]\[
\log 48 \approx 1.68
\][/tex]
[tex]\[
\log 3 \approx 0.48
\][/tex]
Substituting these values into the formula, we obtain:
[tex]\[
\log_3 48 = \frac{\log 48}{\log 3} \approx \frac{1.68}{0.48}
\][/tex]
Now, we perform the division:
[tex]\[
\frac{1.68}{0.48} \approx 3.5
\][/tex]
Therefore, the approximate value of [tex]\(\log_3 48\)[/tex] is:
[tex]\[
\boxed{3.5}
\][/tex]
