Question

Log₂ (x²-100) - log₂ (x + 10) = 1

Answer 1

x = 12

Explanation:

To find the value of x, use logarithmic rules to solve the given equation.

Given logarithmic equation:

`\log_2 (x^2-100) - \log_2 (x + 10) = 1`

`\textsf{Apply the log quotient law:} \quad \log_ax - \log_ay=\log_a \left((x)/(y)\right)`

`\implies \log_2 \left((x^2-100)/(x+10)\right) = 1`

Factor the numerator x² - 100:

`\implies \log_2 \left(((x-10)(x+10))/(x+10)\right) = 1`

Cancel the common factor (x + 10):

`\implies \log_2 \left(x-10\right) = 1`

`\textsf{Apply the log law:} \quad \log_ab=c \iff a^c=b`

`\implies 2^1=x-10`

Simplify and solve for x:

`\implies 2=x-10`

`\implies 2+10=x-10+10`

`\implies 12=x`