Problem 1: Solve for x in the equation

Answer: is the exact answer and x=104.142857143 is an approximate answer.Solution: Step 1: Since you cannot take the log of a negative number, we have to restrict the domain so that 7x >0 or x > 0. Step 2: Isolate the Log term in the original equation by subtracting 4 from each side of the equation:

Step 3: Convert the above logarithmic equation to an exponential equation with base 3 and exponent 6:

Step 4: Divide both sides of the above equation by 7:

is the exact answer and is an approximate answer. Check: Let's substitute the approximate value in the answer and determine whether the left side of the equation equals the right side of the equation after the substitution. Remember we rounded the number and the answer is only a close approximation, so the left and right side of the equation will most likely be very close but not equal; it depends on the number of decimals were rounded in your answer

or

Since the value of the left side of the equation is very close to 10 when you substitute the value of x, and the right side of the equal is 10, you have proved your answer. It won't check exactly because we rounded the value of x.Problem 2: Solve for x in the equation

Answer: is the exact answer and x=10.5427684616 is an approximate answer.Solution: Step 1: Since you cannot take the log of a negative number, we must restrict the domain (values of x) so that

Step 2: Convert the original equation to an exponential equation with base e and exponent 3:

Step 3: Add 1 to both sides of the above equation:

Step 4: Divide both sides of the above equation by 2:

is the exact answer and is an approximate answer. Check: Let's substitute the approximate value of the answer in the original equation and determine whether the left side of the equation equals the right side of the equation after the substitution. Remember we rounded the answer; therefore, the left and right sides of the equation will most likely be very close but may not equal

Since the value of the left side of the equation is 3 when you substitute x = 10.5427684616, and the right side of the equal is 3, you have proved your answer.Problem 3: Solve for x in the equation

Answer: x=e-1 is the exact answer and x=1.71828182846 is an approximate answer.Solution: Step 1: Simplify the left side of the equation using Logarithmic Rule 3:

Step 2: can be simplified to only if we restrict the domain (values of x) so that the quantity x+1>0 or x>-1. Recall that we have to make this restriction on the domain because you can only take the logarithm of a positive number. Step 3: Isolate the Ln term by dividing both sides by 2:

Step 4: Convert the logarithmic equation to an exponential equation with base e and exponent 1:

is the exact answer and

is an approximate answer. Check: Let's substitute the approximate value x=1.71828182846 in the original equation and determine whether the left side of the equation equals the right side of the equation after the substitution. Remember we rounded the number and it is only a close approximation. Therefore, the left and right sides of the equation will most likely be very close but may not equal; it depends on the number of decimals you kept in the rounding process.

Since the value of the left side of the equation is 2 when you substitute the value of x, and the right side of the equations is 2, you have proved your answer.Problem 4: Solve for x in the equation

Answer: x = 2Solution: As you know by now, we can only take the logarithm of a positive number. Therefore, we will have to restrict the domain (values of x) so that the original equation is valid. Step 1: The term is valid when x is greater than zero, and the term is valid when x is greater than 1. If we restrict the domain to the set of all real numbers that are greater than 1, both terms will be valid. Step 2: Simplify the left side of the original equation using Logarithmic Rule 2:

Step 3: Convert the logarithmic term to an exponential term with base 4 and exponent :

Step 4: Simplify the left side of the above equation:

Step 5: Multiply both sides of the above equation by (x - 1):

Step 6: Expand the left side of the above equation:

Step 7: Subtract x from both sides of the above equation and add 2 to both sides of the above equation:

Check: Let's substitute the value x = 2 in the original equation and determine whether the left side of that equation equals the right side of that equation after the substitution

Since the value of the left side of the equation is after you substitute the value 2 for x, and the right side of the equal is , you have proved your answer.Problem 5: Solve for x in the equation

Answer: is the exact answer and x-0.7639320225 is an approximate answer.Solution: Step 1: As we know by now, we can only take the logarithm of a positive number. Therefore, we need to make a restriction on the domain (values of x) so that the problem will be valid (have an answer).The term is valid when x + 5 >0 or x > -5; the term is valid when x + 2 > 0 or x > -2; and the term is valid when x + 6 > 0 or x > -6. If we require that the domain be restricted to the set of all real numbers such that x > -2, all the terms will be valid. Step 2: Simplify the left side of the original equation using Logarithmic Rule 1:

Step 3: We now have an equation of the form which implies that the expression a must equal the expression b or:

Step 4: You could also raise the base 2 to an exponent equal to the left side of the equation in Step 2, and you could raise the base 2 to an exponent equal to right side of the equation in Step 2:

When the base is the same as the base of the logarithm, the above equation can be simplified to

Step 5: Simplify the left side of the above equation:

Step 6: Subtract x and subtract 6 from both sides of the above equation:

Step 7: Use the quadratic formula to solve for x:

There are two exact answer: and and there are two approximate answers:

However only one of the answers is valid.One of the answers ( ) is out of our specified domain of the set of all real numbers such that (x > -2).Therefore, the exact and approximate answers are:

Check: Let's substitute the value in the original equation and determine whether the left side of the equation equals the right side of the equation after the substitution. In other words, does

or

or

or

or

The answer checks. We can also check our solution with the approximate answer. Does

or does

In another word,

Since the value of the left side of the original equation equals the right side of the original equation when we substitute the exact and the approximate value of x, We have proved the answer.Let's illustrate why we had to throw out one of the answers. Let's check to see if the approximate solution (the one we discarded) works:

At this point we have to stop because we cannot take the log of a negative number. We simply cannot calculate the value if any of the terms are undefined.We want to find the solutions to log(8x) - log(1 + ) = 2. Let us note that the equation is only defined when the input for the logarithms is positive. Thus we need x > 0 and 1 + > 0. Since the second condition is automatically satisfied when x > 0, the equation is defined for all positive x. We begin by combining the two logarithmic expressions into one expression, using the rule that log = loga - logb. Consequently our equation becomes log = 2. This is equivalent to saying that = 102 = 100 Next we multiply both sides by the denominator on the left: 8x = 100(1 + ), or equivalently 8x - 100 = 100. Next we square both sides to eliminate the square root term: (8x - 100)2 = 10, 000x 64x2 - 1, 600x + 10, 000 = 10, 000x. Simplifying we obtain the quadratic equation 8x2 - 1, 450x + 1, 250 = 0. Using the quadratic formula, we compute two solutions to this quadratic equation as x = . Numerically the solutions equal x1 180.383 and x2 0.86621. It is essential that you now check both answers in the original equation! For x1, log(8x1) - log(1 + ) 2.000, thus x1 is a solution to the equation. On the other hand, log(8x2) - log(1 + ) .555, thus x2 is not a solution to the equation. Answer: The equation has exactly one solution, namely x = 180.383.