Use the values [tex]\log 48 \approx 1.68[/tex] and [tex]\log 3 \approx 0.48[/tex] to find the approximate value of [tex]\log _3 48[/tex].

[tex]\log _3 48 \approx \qquad[/tex]

The solution is [tex]\square[/tex]



Answer :

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]