15^(-x-3)=-17^(-3x) using a log base 10
To solve the equation 15^(-x-3) = -17^(-3x) using a logarithm with base 10, we can first take the logarithm of both sides of the equation. Since you specified using log base 10, we will use that here:
log₁₀(15^(-x-3)) = log₁₀(-17^(-3x))
Next, we can use the logarithmic property that states logₐ(b^c) = c * logₐ(b). Applying this property, we get:
(-x-3) * log₁₀(15) = -3x * log₁₀(-17)
Now, we need to solve for x. Let's simplify the equation step by step:
1. Distribute the logarithm on both sides:
-x * log₁₀(15) - 3 * log₁₀(15) = -3x * log₁₀(-17)
2. Move all terms containing x to one side of the equation:
-3 * log₁₀(15) = -3x * log₁₀(-17) + x * log₁₀(15)
3. Factor out x on the right side:
-3 * log₁₀(15) = x * (log₁₀(15) - 3 * log₁₀(-17))
4. Divide both sides by (log₁₀(15) - 3 * log₁₀(-17)) to isolate x:
x = -3 * log₁₀(15) / (log₁₀(15) - 3 * log₁₀(-17))
Now you can use a calculator to find the numerical value of x by substituting the values of log₁₀(15) and log₁₀(-17) into the equation.