Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log2 (x2 - 100) - log2 (x + 10) = 1

Accepted Answer

Recall the logarithmic property that states $$ \log_b A - \log_b B = \log_b \left( \frac{A}{B} ight) . $$Apply this to combine the left side: $$ \log_2 \left( \frac{x^2 - 100}{x + 10} ight) = 1 . $$Simplify the expression inside the logarithm. Notice that $$ x^2 - 100  is a $$difference of squares and can be factored as $$ (x - 10)(x + 10) . $$Substitute this to get $$ \log_2 \left( \frac{(x - 10)(x + 10)}{x + 10} ight) = 1 . $$Cancel the common factor $$ x + 10  in $$the numerator and denominator, assuming $$ x + 10 
eq 0 $$, to simplify the logarithm to $$ \log_2 (x - 10) = 1 . $$Rewrite the logarithmic equation in its equivalent exponential form: $$ 2^1 = x - 10 . $$This means $$ 2 = x - 10 . $$Solve the resulting linear equation for $$ x  by $$adding 10 to both sides: $$ x = 2 + 10 . $$Also, check the domain restrictions to ensure the solution makes the original logarithmic expressions defined (arguments must be positive).

Solve each equation. Give solutions in exact form. See Examples 5–9. log₂ (x² - 100) - log₂ (x + 10) = 1

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Key Concepts

Properties of Logarithms

Solving Logarithmic Equations

Domain Restrictions in Logarithms