Answer/Discussion on Logarithmic Equations

(Back to the tutorial on logarithmic equations)

College Algebra
Answer/Discussion to Practice Problems
on Logarithmic Equations

Answer/Discussion to 1a

Step 1: Write as one log isolated on one side.

This is already done for us in this problem.

Step 2: Use the definition of logarithms to write in exponential form.

*Rewrite in exponential form
*Base = 10 and exponent = 3

Remember that when there is no base written on a log, that means it is the common log, or log base 10. If you need a review on common logs feel free to go to Tutorial 43: Logarithmic Functions.

Step 3: Solve for x.

*10 cubed is 1000
*Solve for x

(return to problem 1a)

Answer/Discussion to 1b

Step 1: Write as one log isolated on one side.

*Use the product rule to write as one log
*one log is isolated

Step 2: Use the definition of logarithms to write in exponential form.

*Rewrite in exponential form
*Base = 2 and exponent = 4

Step 3: Solve for x.

*2 to the 4th power is 16
*Solve for x

*Factor the trinomial

*Set 1st factor = 0

*Set 2nd factor = 0

Since -2 would create a negative number inside both logs in this problem and we CANNOT take the log of a negative number, we will have to throw out x = -2 as one of our solutions.

Final answer: x = 8.

(return to problem 1b)

Answer/Discussion to 1c

Step 1: Write as one log isolated on one side.

*Use the quotient rule to write as one log
*one log is isolated

Step 2: Use the definition of logarithms to write in exponential form.

*Rewrite in exponential form
*Base = 4 and exponent = 2

Step 3: Solve for x.

*4 squared is 16
*Solve for x

*Multiply both sides by LCD of x - 1

The equation that we had to solve in step 3 had a rational expression in it. If you need a review on solving equations with rational expressions feel free to go to Tutorial 15: Equations with Rational Expressions.

(return to problem 1c)

Answer/Discussion to 1d

Step 1: Write as one log isolated on one side.

*Inverse of add 5 is sub. 5

*one log is isolated

Step 2: Use the definition of logarithms to write in exponential form.

*Rewrite in exponential form
*Base = e and exponent = 1

Remember that when you have ln, that means it is the natural log, or log base e. If you need a review on natural logs feel free to go to Tutorial 43: Logarithmic Functions.

Step 3: Solve for x.

*Square both sides to get rid of the radical

*Solve for x

*Use the calculator to find e squared

The equation that we had to solve in step 3 had a square root in it. If you need a review on solving equations with radicals feel free to go to Tutorial 19: Radical Equations and Equations Involving Rational Exponents.

(return to problem 1d)

(Back to the tutorial on logarithmic equations)