3 Ways to Solve Logarithms

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

Logarithms can be intimidating, but solving a logarithm is much easier once you realize that logarithms are just a different way to write exponential equations. Once the logarithms are rewritten in a more familiar form, you should be able to solve them as a standard exponential equation.

Steps

Learn to Express Logarithmic Equations Exponentially

Step 1. Learn the Definition of Logarithm

Before you can solve logarithms, you need to understand that a logarithm is essentially a different way to write exponential equations. Its precise definition is as follows:

• y = log_b (x)

  If and only if: b^y = x

• Note that b is the base of the logarithm. It must also be true that:

  • b> 0
  • b is not equal to 1
• In the same equation, y is the exponent and x is the exponential expression to which the logarithm is equaled.

Step 2. Analyze the equation

When you are faced with a logarithmic problem, identify the base (b), the exponent (y), and the exponential expression (x).

• Example:

  5 = log₄(1024)

  • b = 4
  • y = 5
  • x = 1024

  

  Step 3. Move the exponential expression to one side of the equation

  Place the value of your exponential expression, x, on one side of the equal sign.

  • Example: 1024 = ?

    

    Step 4. Apply the exponent to the base

    The value of your base, b, must be multiplied by itself the number of times indicated by the exponent, y.

    • Example:

      4 * 4 * 4 * 4 * 4 = ?

      This could also be written as: 4⁵

      

      Step 5. Rewrite your final answer

      You should now be able to rewrite your logarithm as an exponential expression. Check that your expression is correct by making sure that the members on both sides of the equals are equivalent.

      Example: 4⁵ = 1024

      Method 1 of 3: Method 1: Solve for X

      

      Step 1. Isolate the logarithm

      Use the inverse operation to bring all the parts that are not logarimic to the other side of the equation.

      • Example:

        log₃(x + 5) + 6 = 10

        • log₃(x + 5) + 6 - 6 = 10 - 6
        • log₃(x + 5) = 4

        

        Step 2. Rewrite the equation in exponential form

        Using what you know about the relationship between logarithmic equations and exponentials, break down the logarithm and rewrite the equation in exponential form, which is easier to solve.

        • Example:

          log₃(x + 5) = 4

          • Comparing this equation with the definition [ y = log_b (x)], you can conclude that: y = 4; b = 3; x = x + 5
          • Rewrite the equation so that: b^y = x
          • 3⁴ = x + 5

          

          Step 3. Solve for x

          With the simplified problem to an exponential, you should be able to solve it as you would solve an exponential.

          • Example:

            3⁴ = x + 5

            • 3 * 3 * 3 * 3 = x + 5
            • 81 = x + 5
            • 81 - 5 = x + 5 - 5
            • 76 = x

            

            Step 4. Write your final answer

            The solution you find solving for x is the solution of your original logarithm.

            • Example:

              x = 76

            Method 2 of 3: Method 2: Solve for X Using the Logarithmic Product Rule

            

            Step 1. Learn the product rule

            The first property of logarithms, called the "product rule," says that the logarithm of a product is the sum of the logarithms of the various factors. Writing it through an equation:

            • log_b(m * n) = log_b(m) + log_b(n)

            • Also note that the following conditions must be met:

              • m> 0
              • n> 0

              

              Step 2. Isolate the logarithm from one side of the equation

              Use the operations of the inverai to bring all the parts containing logarithms on one side of the equation and all the rest on the other.

              • Example:

                log₄(x + 6) = 2 - log₄(x)

                • log₄(x + 6) + log₄(x) = 2 - log₄(x) + log₄(x)
                • log₄(x + 6) + log₄(x) = 2

                

                Step 3. Apply the product rule

                If there are two logarithms that are added together within the equation, you can use the logarithm rules to combine them together and transform them into one. Note that this rule only applies if the two logarithms have the same base

                • Example:

                  log₄(x + 6) + log₄(x) = 2

                  • log₄[(x + 6) * x] = 2
                  • log₄(x² + 6x) = 2

                  

                  Step 4. Rewrite the equation in exponential form

                  Remember that the logarithm is just another way to write the exponential. Rewrite the equation in a solvable form

                  • Example:

                    log₄(x² + 6x) = 2

                    • Compare this equation with the definition [ y = log_b (x)], then conclude that: y = 2; b = 4; x = x² + 6x
                    • Rewrite the equation so that: b^y = x
                    • 4² = x² + 6x

                    

                    Step 5. Solve for x

                    Now that the equation has become a standard exponential, use your knowledge of exponential equations to solve for x as you normally would.

                    • Example:

                      4² = x² + 6x

                      • 4 * 4 = x² + 6x
                      • 16 = x² + 6x
                      • 16 - 16 = x² + 6x - 16
                      • 0 = x² + 6x - 16
                      • 0 = (x - 2) * (x + 8)
                      • x = 2; x = -8

                      

                      Step 6. Write your answer

                      At this point you should know the solution of the equation, which corresponds to that of the starting equation.

                      • Example:

                        x = 2

                      • Note that you cannot have a negative solution for logarithms, so you discard the solution x = - 8.

                      Method 3 of 3: Method 3: Solve for X Using the Logarithmic Quotient Rule

                      

                      Step 1. Learn the quotient rule

                      According to the second property of logarithms, called the "quotient rule," the logarithm of a quotient can be rewritten as the difference between the logarithm of the numerator and the logarithm of the denominator. Writing it as an equation:

                      • log_b(m / n) = log_b(m) - log_b(n)

                      • Also note that the following conditions must be met:

                        • m> 0
                        • n> 0

                        

                        Step 2. Isolate the logarithm from one side of the equation

                        Before you can solve the logarithm, you have to move all the logarithms to one side of the equation. Everything else should be moved to the other member. Use inverse operations to accomplish this.

                        • Example:

                          log₃(x + 6) = 2 + log₃(x - 2)

                          • log₃(x + 6) - log₃(x - 2) = 2 + log₃(x - 2) - log₃(x - 2)
                          • log₃(x + 6) - log₃(x - 2) = 2

                          

                          Step 3. Apply the quotient rule

                          If there is a difference between two logarithms having the same base within the equation, you must use the rule of quotients to rewrite the logarithms as one.

                          • Example:

                            log₃(x + 6) - log₃(x - 2) = 2

                            log₃[(x + 6) / (x - 2)] = 2

                            

                            Step 4. Rewrite the equation in exponential form

                            Remember that the logarithm is just another way to write the exponential. Rewrite the equation in a solvable form.

                            • Example:

                              log₃[(x + 6) / (x - 2)] = 2

                              • Comparing this equation to the definition [ y = log_b (x)], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2)
                              • Rewrite the equation so that: b^y = x
                              • 3² = (x + 6) / (x - 2)

                              

                              Step 5. Solve for x

                              With the equation now in exponential form, you should be able to solve for x as you normally would.

                              • Example:

                                3² = (x + 6) / (x - 2)

                                • 3 * 3 = (x + 6) / (x - 2)
                                • 9 = (x + 6) / (x - 2)
                                • 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2)
                                • 9x - 18 = x + 6
                                • 9x - x - 18 + 18 = x - x + 6 + 18
                                • 8x = 24
                                • 8x / 8 = 24/8
                                • x = 3

                                

                                Step 6. Write your final solution

                                Go back and double check your steps. Once you are sure you have the correct solution, write it down.

                                • Example:

                                  x = 3