Solve for x.
14^(-3x)=11^(-x-3)
Write the exact answer using base-10 logarithms
How do I start this problem? Thank You
To solve for x in the equation 14^(-3x) = 11^(-x-3), you can start by taking the logarithm of both sides of the equation using base-10 logarithms. Here are the steps to solve:
Step 1: Take the logarithm of both sides using base-10 logarithms. This will help us simplify the equation:
log (14^(-3x)) = log (11^(-x-3))
Step 2: Apply the exponent rule of logarithms, which states that log (a^b) = b * log (a):
(-3x) * log (14) = (-x - 3) * log (11)
Step 3: Distribute the logarithm on the right side of the equation:
-3x * log (14) = (-x * log (11)) - (3 * log (11))
Step 4: Move all terms containing x to one side and constants to the other side to simplify the equation:
-3x * log (14) + x * log (11) = -3 * log (11)
Step 5: Factor out x from the left side of the equation:
x * (-3 * log (14) + log (11)) = -3 * log (11)
Step 6: Divide both sides of the equation by the coefficient of x to solve for x:
x = (-3 * log (11)) / (-3 * log (14) + log (11))
Therefore, the exact solution for x using base-10 logarithms is:
x = (-3 * log (11)) / (-3 * log (14) + log (11))
To solve for x in this equation, you can start by taking the logarithm of both sides of the equation. In this case, you can use the base-10 logarithm (log) to simplify the equation.
Here's how you can start:
1. Take the logarithm of both sides of the equation using base-10 logarithm (log):
log(14^(-3x)) = log(11^(-x-3))
2. Apply the logarithm property:
(-3x)log(14) = (-x-3)log(11)
3. Expand the equation using the logarithm property (log(a^b) = b * log(a)):
-3x * log(14) = -x * log(11) - 3 * log(11)
Now, you can continue simplifying the equation to solve for x.