solve: log8(w-6)=2-log8(w+15)
note: the 8 is to be little at the bottom right corner of log
To solve the given equation, we will use logarithmic properties and algebraic manipulations. Here's a step-by-step solution:
Step 1: Apply the logarithmic property: log a - log b = log (a / b).
log8(w - 6) = 2 - log8(w + 15)
log8(w - 6) + log8(w + 15) = 2
Step 2: Combine the logarithms on the left side using the logarithmic property: log a + log b = log (a * b).
log8[(w - 6)(w + 15)] = 2
Step 3: Rewrite this equation in exponential form. Recall that logarithms are exponents, so if y = log base a of x, then x = a^y.
8^2 = (w - 6)(w + 15)
64 = (w - 6)(w + 15)
Step 4: Simplify the equation.
64 = w^2 + 15w - 6w - 90
64 = w^2 + 9w - 90
Step 5: Arrange the equation in standard quadratic form (ax^2 + bx + c = 0).
w^2 + 9w - 90 - 64 = 0
w^2 + 9w - 154 = 0
Step 6: Solve the quadratic equation. We can either factor it or use the quadratic formula.
Factoring:
(w - 7)(w + 22) = 0
Setting each factor equal to zero:
w - 7 = 0 or w + 22 = 0
Solving for w:
w = 7 or w = -22
So the solutions to the equation log8(w - 6) = 2 - log8(w + 15) are w = 7 or w = -22.