Find the answer to the following equation: 2log(x-4) - 5 = 9.
To solve a logarithmic equation, you need to isolate the logarithm itself. In this particular problem, the 'log' is attached to the 'x-4', so you'd try to get rid of the 2 being multiplied to the log and the 5 being subtracted from it. Now, if the problem stated 2x - 5 = 9, you'd likely have no issues solving the problem because you'd simply add 5 to both sides and then divide both sides by 2. The same applies to logarithm problems. To solve 2log(x-4) - 5 = 9, you must first add 5 to both sides. This gives you: 2log(x-4) = 14 Then you'd dividie by 2. That would give you: log(x-4) = 7 Now the logarithm is completely isolated. Unfortunately, we need to get x by itself, but we can't directly separated it from the logarithm. The 'log' is married to the 'x-4'. Instead, you must use the defition of logarithm to finish solving the problem, and you must rewrite the logarithm as an exponential problem. In this problem, no base is written on the logarithm, so the base is 10 by default. 10 will be the base of the exponential problem, 7 will be the exponent, and x-4 will be the product. In other words, you can re-write log(x-4) = 7 as 10^7 = x - 4. From here, solving for x is simple! Just add 4 to both sides. The answer will be 10^7 + 4.
Find all solutions on the interval [0, 2pi] for the equation sin2x = cosx.
This problem involves a double-angle formula because we see that sin2x involves two times an angle. Now, the formula for sin2x is 2sinx cosx. It's impossible to solve the equation as it currently stands, due to the angles being different. If we replace sin2x with 2sinx cosx however, we will then be dealing with one angle throughout the entire problem. So our first step would be to replace sin2x, and the result would look like this: 2sinx cosx = cosx. Since there are now two trigonometric functions (sinx AND cosx), and nothing can be written in simpler terms, it'd be a good idea to set the equation =0. This means we must move everything to one side. We can do this by subtracting cosx from both sides of the equal sign. This results in the following equation: 2sinx cosx - cosx = 0 Note that these two terms cannot be combined. It'd be similar to combining xy with x... it's not possible because they aren't like terms. However, both terms DO share cosx, which is the GCF. Thus, it'd be a good idea to factor cosx out from the equation. Factoring cosx out results in the following: cosx (2sinx - 1) = 0 (You can check this by distributing everything back out!) Now that our problems if factored, we can use the zero product rule (anything times 0 is 0) to break this up into two smaller problems. Those problems would be: cosx = 0 and 2sinx - 1 =0 The first equation already has the trigonometric function by itself, but the second one needs to be worked with. We can isolate sinx by adding 1 to both sides and then dividing by 2. That gives us: cosx =0 and sinx = 1/2 These are asking us for the angles in which cosx is 0 (or where x is 0) and sinx = 1/2 (or where y = 1/2). The places this occurs within the window [0,2pi] are: pi/2, 3pi/2, pi/6, and 5pi/6.
The sum of three consecutive numbers is 111. What is the smallest of the three numbers ?
When solving word problems, it's often a good idea to develope a set of hypothetical questions to ask yourself. For example, in the problem above, we know we are looking for a sum of three consecutive numbers. This means we want three "in-order" numbers that add up to be 111. Let's think of any 3 consecutive numbers and consider their relationships. One example would be 3, 4, and 5. How can we describe these numbers? Ask yourself "How does 4 relate to 3?" The answer is that 4 is one more than 3... or 3+1. Likewise, 5 is 3 + 2. Now, we have no idea what our three numbers are. Whenever we don't know the value of a number, we assign it a variable. Let's say our first number is 'x'. We want the next two numbers that come after x. The number after x would be 'x+1' (Remember how 4 is 3+1?) and the last number would be 'x+2' (Just like 5 is 3+2). The problem asks for us to consider the sum of our three numbers. In other words, we must add x, x+1, and x+2. That looks like this: x + x+1 + x+2. We can collect like terms. x + x+1 + x+2 = 3x + 3. We also know that this sum... 3x + 3 ... should add up to be 111. In other words, 3x + 3 = 111. We can solve this! 3x+3=111 Subtract 3 from both sides. 3x = 108 And divide 3 from both sides. x = 36. Now we know the value of x, which was our smallest number!