How to Solve Logarithms With Different Bases


Logarithms are an important concept for the science and engineering world. A logarithm is the inverse of an exponent, much the same way addition is the inverse of subtraction. Logarithms provide an intuitive means of understanding multiplication by enabling a means of multiplying numbers using addition. Logarithms have a base, which is the number that is raised to some power for exponents. There are many operations that can be performed on logarithms; however, this requires that the logarithms have the same base. Solving logarithms with different bases require a change of base of the logarithms, which can be performed in a few short steps.

  • Write the question you are trying to solve. As an example, assume you are trying to solve the problem: log4(x + 1) + log16(x + 1) = log4(8). In this problem, there are two different bases: 4 and 16.

  • Use the change of base formula to give each term the same base. The change of base formula says that to change the base of logb(x), where b is the base and x is an arbitrary number, rewrite the logarithm as logk(x) / logk(b), where k is an arbitrary number selected as the new base. In the example above, you can change the base of the term log16(x + 1) by rewriting the number as log4(x + 1) / log4(16). This simplifes to log4(x + 1) / 2.

  • Use the rules of logarithms to simplify the problem into solvable form. In the example above, the equation log4(x + 1) + log4(x + 1) / 2 = log4(8) can be simplified to log4(x + 1) + log4(x + 1)^(1/2) = log4(8), using the power rule for logarithms. By using the product rule for logarithms, the equation can be further simplified to log4(x + 1)(x + 1)^(1/2) = log4(8).

  • Eliminate the logarithm. By taking both sides of the equation to the power of 4, the example equation simplifies to (x + 1)(x + 1)^(1/2) = 8, which further simplifies to (x + 1)^(3/2) = 8.

  • Solve for x. In the example above, this is done by taking both sides of the equation to the power of 2/3. This renders x + 1 = 4 and solving for x produces x = 3.


  • Photo Credit Jupiterimages/Goodshoot/Getty Images
Promoted By Zergnet


You May Also Like

Related Searches

Check It Out

Can You Take Advantage Of Student Loan Forgiveness?

Is DIY in your DNA? Become part of our maker community.
Submit Your Work!