Solve the equation.
-$\log _ { 21 } ( x + 84 ) = 3 - \log _ { 21 } x$
A) $\{ 63 \}$
B) $\{ 147 \}$
C) $\{ - 147 \}$
D) $\{ - 63 \}$

Question

Quizplus · Accepted Answer

To solve the equation $\log _ { 21 } ( x + 84 ) = 3 - \log _ { 21 } x$, we can use the properties of logarithms to combine the terms. First, rewrite the equation as $\log _ { 21 } ( x + 84 ) + \log _ { 21 } x = 3$. Using the property that $\log_a(b) + \log_a(c) = \log_a(bc)$, we get $\log _ { 21 } ( x(x + 84) ) = 3$. Converting from logarithmic form to exponential form gives $x(x + 84) = 21^3$. Simplifying, we get $x^2 + 84x = 9261$. Subtracting 9261 from both sides gives $x^2 + 84x - 9261 = 0$. Factoring or using the quadratic formula, we find that $x = 63$ is the solution that makes sense in the context of the original equation (since $x = -147$ would result in taking the logarithm of a negative number, which is undefined).

Solve the Equation log⁡21(x+84)=3−log⁡21x\log _ { 21 } ( x + 84 ) = 3 - \log _ { 21 } xlog21​(x+84)=3−log21​x

Solve the Equation $\log _ { 21 } ( x + 84 ) = 3 - \log _ { 21 } x$