Sure! Let's solve the equation [tex]$\log_3(3z + 4) - \log_3 z = 2$[/tex] step-by-step.
1. Use the logarithmic property: Recall that the difference of logarithms can be written as the logarithm of a quotient:
[tex]\[
\log_3(3z + 4) - \log_3 z = \log_3 \left( \frac{3z + 4}{z} \right)
\][/tex]
2. Set up the new logarithmic equation: Substitute the quotient into the logarithmic function:
[tex]\[
\log_3 \left( \frac{3z + 4}{z} \right) = 2
\][/tex]
3. Convert the logarithmic equation to an exponential equation: Remember that if [tex]$\log_b(x) = y$[/tex], then [tex]$b^y = x$[/tex]. Here, [tex]$b = 3$[/tex] and [tex]$y = 2$[/tex], so:
[tex]\[
\frac{3z + 4}{z} = 3^2
\][/tex]
4. Simplify the exponential equation: Calculate the right-hand side of the equation:
[tex]\[
\frac{3z + 4}{z} = 9
\][/tex]
5. Isolate z: Start by multiplying both sides of the equation by [tex]$z$[/tex]:
[tex]\[
3z + 4 = 9z
\][/tex]
6. Solve for z: Rearrange the equation to isolate [tex]$z$[/tex] on one side:
[tex]\[
3z + 4 - 9z = 0
\][/tex]
[tex]\[
-6z + 4 = 0
\][/tex]
[tex]\[
-6z = -4
\][/tex]
[tex]\[
z = \frac{-4}{-6}
\][/tex]
Simplify the fraction:
[tex]\[
z = \frac{2}{3}
\][/tex]
So, the value of [tex]$z$[/tex] that satisfies the equation [tex]$\log_3(3z + 4) - \log_3 z = 2$[/tex] is [tex]$\boxed{\frac{2}{3}}$[/tex].
