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

Question

Use the properties of logarithms to solve the following equation.

\log_{7} (x + 2) = \log_{7} (14) - \log_{7} (x - 3)

Answer 1

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
$x^{2} + 2 x - 3 x - 6 = 14$
$x^{2} - x - 20 = 0$
Step 3
Solve the quadratic equation as follows:
$x^{2} - 5 x + 4 x - 20 = 0$
x(x-5)+4(x-5)=0
(x-5)(x+4)=0
x=5, -4
Hence the values of x are 5 and −4.

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

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