To solve the equation [tex]\(\log_2(3t + 4) - \log_2 t = 3\)[/tex], follow these steps:
1. Use Logarithm Properties:
Recall the logarithm property:
[tex]\[
\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)
\][/tex]
Apply this property to our equation:
[tex]\[
\log_2 \left(\frac{3t + 4}{t}\right) = 3
\][/tex]
2. Simplify the Fraction:
Simplify the argument of the logarithm:
[tex]\[
\frac{3t + 4}{t} = \frac{3t}{t} + \frac{4}{t} = 3 + \frac{4}{t}
\][/tex]
Rewrite the equation:
[tex]\[
\log_2 \left(3 + \frac{4}{t}\right) = 3
\][/tex]
3. Eliminate the Logarithm:
To isolate [tex]\(t\)[/tex], rewrite the logarithmic equation in its exponential form. Recall that if [tex]\(\log_b (A) = C\)[/tex], then [tex]\(A = b^C\)[/tex]. Hence:
[tex]\[
3 + \frac{4}{t} = 2^3
\][/tex]
Since [tex]\(2^3 = 8\)[/tex]:
[tex]\[
3 + \frac{4}{t} = 8
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
Isolate [tex]\(\frac{4}{t}\)[/tex] by subtracting 3 from both sides:
[tex]\[
\frac{4}{t} = 5
\][/tex]
Solve for [tex]\(t\)[/tex] by multiplying both sides by [tex]\(t\)[/tex] and then dividing by 5:
[tex]\[
4 = 5t
\][/tex]
[tex]\[
t = \frac{4}{5}
\][/tex]
Therefore, the solution to the equation [tex]\(\log_2(3t + 4) - \log_2 t = 3\)[/tex] is:
[tex]\[
t = \frac{4}{5}
\][/tex]
