1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable. Example 1: Solve for x in the equation Ln(x)=8. Solution: Step 1: Let both sides be exponents of the base e. The equation Ln(x)=8 can be rewritten
and the approximate answer is
Check:
You can check your answer in two ways. You could graph the function Ln(x)-8 and see where it crosses the x-axis. If you are correct, the graph should cross the x-axis at the answer you derived algebraically. Example 2: Solve
for x in the equation 7Log(3x)=15. Solution: Step 1: Isolate the logarithmic term before you convert the logarithmic equation to an exponential equation. Divide both sides of the original equation by 7:
Step 2: Convert the logarithmic equation to an exponential equation: If no base is indicated, it means the base of the logarithm is 10. Recall also that logarithms are exponents, so the exponent is
can now be written
Check: You can check your answer in two ways: graphing the function
or substituting the value of x into the original equation. If you choose graphing, the x-intercept should be the same as the answer you derived ( If you choose substitution, the value of the left side of the original equation should equal the value of the right side of the equation after you have calculated the value of each side based on your answer for x. Example 3: Solve for x in the equation
Solution: Step 1: Note the first term Ln(x-3) is valid only when x>3; the term Ln(x-2) is valid only when x>2; and the term Ln(2x+24) is valid only when x>-12. If we require that x be any real number greater than 3, all three terms will be valid. If all three terms are valid, then the equation is valid. Step 2: Simplify the left side of the above equation: By the properties of logarithms, we know that
Step 3: The equation can now be written
Step 4: Let each side of the above equation be the exponent of the base e:
Another way of looking at the equation in Step 3 is to realize that if Ln(a) = Ln(b), then a must equal b. In the case of this problem, then
Step 6: Simplify the left side of the above equation:
Step 7: Subtract 2x + 24 from each side:
Step 8: Factor the left side of the above equation:
Step 9: If the product of two factors equals zero, at least one of the factor has to be zero. If Check: You can check your answer by graphing the function
and determining whether the x-intercept is also equal to 9. If it is, you have worked the problem correctly. If you would like to review another example, click on Example. Work the following problems. If you wish to review the answer and the solution, click on Answer. Problem 1: Solve for x in the equation
Answer Problem 2: Solve for x in the equation
Answer Problem 3: Solve for x in the equation
Answer Problem 4: Solve for x in the equation
Answer Problem 5: Solve for x in the equation
Answer Problem 6: Solve for x in the equation
Answer [Back to Rules of Logarithms] [Back to Exponential Functions] [Algebra] [Trigonometry] [Complex Variables] Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Copyright � 1999-2022 MathMedics, LLC. All rights reserved. What is the first step to solving a logarithmic equation?Step 1: The first step in solving a logarithmic equation is to isolate the logarithmic term on one side of the equation. Our equation log 7 (x – 3) = 17 is already in this form so we can move on to the next step.
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