Tutor profile: Jackie K.
Use the properties of logarithms to simplify the following expression: log[((x-2)(x+5))/(x-4)^3]
The three big properties we need to use here are: 1) log(xy) = log(x) + log(y) 2) log(x/y) = log(x) - log(y) 3) log(x^r) = rlog(x) So, let's first use Property 2 to write: log[(x-2)(x+5)] - log[(x-4)^3]. Now, let's use Property 3 to simplify the second term and write: log[(x-2)(x+5)] - 3log(x-4). Lastly, let's use Property 1 to finish simplifying the first term: log(x-2) + log(x+5) - 3log(x-4). Therefore, our final answer is log(x-2) + log(x+5) - 3log(x-4).
Find the derivative of f(x) = lnx + 2cosx - 5.
Recall that the formula for the derivative of lnx is 1/x, the derivative of cosx is -sinx, and that the derivative of a constant (like -5) is simply 0. Putting all this information together, we can conclude that f'(x) = 1/x - sinx.
Solve the following equation for x: 2x+3 = -3x+18
When solving an equation for x, the goal is to first get all the "x" terms on one side of the equation and the constant terms on the other, then to simplify the resulting equation in order to find the final answer. So, for this example, let's start by adding "3x" to both sides. This gives us: 5x+3 = 18. Now, let's subtract "3" from both sides. This gives us: 5x = 15. The final step is to divide both sides by "5", which gives us x=3 as our answer.
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