Question

1. $\log_{2}(x - 1)=3$
2. $\log_{5}(2x - 1)=2$
3. $\ln(3 - 4x)=0$
4. $3\log_{4}(x)+2=-7$
5. $4 = \frac{\ln(x + 5)}{3}$
6. $5 - 3\log(3x + 2)=2$

Explanation:

Problem 7: $\boldsymbol{\log_2(x - 1) = 3}$

Step1: Convert log to exponential form

Using the definition of logarithms: if $\log_b(a) = c$, then $b^c = a$. Here, $b = 2$, $c = 3$, and $a = x - 1$. So we get $2^3 = x - 1$.
$$2^3 = x - 1$$

Step2: Simplify and solve for x

Calculate $2^3 = 8$. Then the equation becomes $8 = x - 1$. Add 1 to both sides: $x = 8 + 1 = 9$.

Step1: Convert log to exponential form

Using $\log_b(a) = c \implies b^c = a$. Here, $b = 5$, $c = 2$, $a = 2x - 1$. So $5^2 = 2x - 1$.
$$5^2 = 2x - 1$$

Step2: Simplify and solve for x

$5^2 = 25$, so $25 = 2x - 1$. Add 1 to both sides: $26 = 2x$. Divide by 2: $x = 13$.

Step1: Convert natural log to exponential form

Recall that $\ln(a) = c$ is equivalent to $e^c = a$. Here, $c = 0$, so $e^0 = 3 - 4x$.
$$e^0 = 3 - 4x$$

Step2: Simplify and solve for x

Since $e^0 = 1$, we have $1 = 3 - 4x$. Subtract 3: $-2 = -4x$. Divide by -4: $x = \frac{-2}{-4} = \frac{1}{2}$.

Answer:

$x = 9$

Problem 8: $\boldsymbol{\log_5(2x - 1) = 2}$