Question

# Use the properties of logarithms to solve the following equation. \log_{7}(x+2)=\log_{7}(14)−\log_{7}(x−3)

Equations
Use the properties of logarithms to solve the following equation.
$$\log_{7}(x+2)=\log_{7}(14)−\log_{7}(x−3)$$

2021-05-09

Step 1
The second question is opinion question and cannot be answered as per bartleby guidelines.
It is given that:
$$\log_{7}(x+2)=\log_{7}(14)−\log_{7}(x−3)$$
use the formula $$\log a−\log b=\log(\frac{a}{b})$$ to get:
$$\log_{7}(x+2)=\log_{7}(\frac{14}{x-3})$$
Step 2
Cancel log from both sides to get:
$$x+2=\frac{14}{x-3}$$
(x+2)(x-3)=14
$$x^{2}+2x-3x-6=14$$
$$x^{2}-x-20=0$$
Step 3
Solve the quadratic equation as follows:
$$x^{2}-5x+4x-20=0$$
x(x-5)+4(x-5)=0
(x-5)(x+4)=0
x=5, -4
Hence the values of x are 5 and −4.